Research ArticleNew Inertial Relaxed CQ Algorithms for Solving Split FeasibilityProblems in Hilbert Spaces

Haiying Li 1 Yulian Wu 1 and Fenghui Wang 2

1College of Mathematics and Information Science Henan Normal University Xinxiang 453007 China2Department of Mathematics Luoyang Normal University Luoyang 471934 China

Correspondence should be addressed to Haiying Li haiyingli2020163com

Received 3 January 2021 Revised 24 January 2021 Accepted 28 January 2021 Published 16 February 2021

Academic Editor Xiaolong Qin

Copyright copy 2021 Haiying Li et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

)e split feasibility problem (SFP) has received much attention due to its various applications in signal processing and image re-construction In this paper we propose two inertial relaxed CQ algorithms for solving the split feasibility problem in real Hilbert spacesaccording to the previous experience of applying inertial technology to the algorithm)ese algorithms involve metric projections ontohalf-spaces and we construct new variable step size which has an exact form and does not need to know a prior information norm ofbounded linear operators Furthermore we also establish weak and strong convergence of the proposed algorithms under certain mildconditions and present a numerical experiment to illustrate the performance of the proposed algorithms

1 Introduction

)e split feasibility problem in finite-dimensional Hilbertspaces was first introduced by Censor and Elfving [1] in1994 for modeling inverse problem that arises from thephase retrievals and in medical image reconstruction [2])e split feasibility problem can also be used to model theintensity-modulated radiation therapy [3]

Let H1 and H2 be two real Hilbert spaces with theinner product langmiddot middotrang and the induced norm middot C and Q

are nonempty closed and convex subsets of real Hilbertspaces H1 and H2 respectively and A is a linear boundedoperator from H1 into H2 )e split feasibility problem(SFP) is formulated as follows find a point x isin H1satisfying

x isin C

Ax isin Q(1)

)e solution set of the problem (SFP) (1) is denoted by Sthat is

S ≔ x isin C Ax isin Q (2)

A very successful method that solves the (SFP) seems tobe the CQ algorithm of Byrne [4] which generates xn1113864 1113865 bythe iterative procedure for any initial guess x1 isin H

xn+1 PC xn minus cAlowast

I minus PQ1113872 1113873Axn1113872 1113873 forallnge 1 (3)

where PC and PQ are the metric projections onto C and Qrespectively Alowast is the adjoint operator of the linear operatorA and the step size c is chosen in the open interval(0 2A2) )e step size selection depends on the operatornorm (or the largest eigenvalue of AlowastA ) which also is not asimple work

)e CQ algorithm (3) for solving the problem (SFP) (1)can be obtained from optimization If we introduce theconvex objective function

f(x) ≔12

I minus PQ1113872 1113873Ax

2 x isin H1 (4)

then the CQ algorithm (3) comes immediately as a specialcase of the gradient-projection algorithm (GPA) since theconvex objective function f is differentiable and has aLipschitz gradient given by

nablaf(x) Alowast

I minus PQ1113872 1113873Ax (5)

HindawiJournal of MathematicsVolume 2021 Article ID 6624509 13 pageshttpsdoiorg10115520216624509

To overcome the computational difficulties many au-thors have constructed the variable step size that does notrequire the norm A see for example [5ndash12] In particularLopez et al [7] introduced a new choice of the variable stepsize sequence τn as follows

τn ≔ρnf xn( 1113857

nablaf xn( 1113857

2 forallnge 1 (6)

where ρn1113864 1113865 is a sequence of positive real numbers take zerofor the lower bound and four for the upper bound )eadvantage of the choice (6) of step size is that there is neitherprior information about the matrix norm A nor any otherconditions on Q and A

Now let us consider the case when C and Q are levelsubsets of convex functions where C and Q are respectivelygiven by

C x isin H1 c(x)le 01113864 1113865

Q y isin H2 q(y)le 01113864 1113865(7)

where c H1⟶ (minus infin +infin] and q H2⟶ (minus infin +infin] aretwo lower semicontinuous convex functions and zc and zq

are bounded operators But the associated projections PC

and PQ do not have closed-form expressions and the CQ

algorithm is that the iterative process cannot be performedIn order to keep it going Yang [13] made improvements tothese two-level subsets here is how they are defined

1113958Cn x isin H1 c xn( 1113857 +langξn x minus xnrang le 01113864 1113865 (8)

with ξn isinzc(xn) and1113958Qn y isin H2 q Axn( 1113857 +langζn y minus Axnrangle 01113864 1113865 (9)

with ζn isinzq(Axn)It is easy to see that 1113958Cn and 1113958Qn are both half-spaces and

the projections P 1113957Cn

and P 1113957Qn

have closed-form expressions In

what follows for each nge 1 define

fn(x) ≔12

I minus P 1113957Qn

1113874 1113875Ax

2

nablafn(x) Alowast

I minus P 1113957Qn

1113874 1113875Ax

(10)

Since these projections are easy to calculate the algo-rithm is very practical

Afterwards the inertial technique was developed byAlvarez and Attouch in order to improve the performance ofproximal point algorithms [14] Dang et al [15] proposed aninertial relaxed CQ algorithm xn1113864 1113865 for solving the problem(SFP) in a real Hilbert space which is generated as followsfor any x0 x1 isin H

wn xn + θn xn minus xnminus 1( 1113857

xn+1 PCn

wn minus cAT

I minus PQn

1113874 1113875A wn( 11138571113874 1113875

⎧⎪⎨

⎪⎩(11)

where 0lt clt (2A2) and 0le θn le θn with

θn min θ1

max n2

xn minus xnminus 1

2 n

2xn minus xnminus 1

2

1113882 1113883

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ forallnge 1 θ isin [0 1)

Cn x isin H1|c wn( 1113857 +langξn x minus wnrang le 01113864 1113865

(12)

with ξn isinzc(wn) and

Qn y isin H2|q Awn( 1113857 +langζn y minus Awnrang le 01113864 1113865 (13)

with ζn isinzq(Awn) )e algorithm xn1113864 1113865 converges weaklyto a point of a solution set of the problem (SFP) wherestep size also depends on the matrix norm A It isobvious that the calculation of operator norm is morecomplicated so Gibali et al [16] has changed the step sizeof (11)

λn ρnfn wn( 1113857

η2n

ηn max 1 nablafn wn( 1113857

1113966 1113967 0le θn le θn

(14)

where

θn

min θεn

xn minus xnminus 1

2

⎧⎨

⎩

⎫⎬

⎭ if xn nexnminus 1

θ otherwise

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(15)

If 1113936infinn1 θnxn minus xnminus 1

2 ltinfin then the sequence xn1113864 1113865

generated by (11) with step size λn converges weakly to apoint of a solution set of the problem (SFP) For recentresults on inertial algorithms (see [17ndash24])

On the other hand the CQ algorithm is the gradient-projection method for the variational inequality problem In[25] Xu gave weak convergence in the setting of Hilbertspaces Wang and Xu [26] proposed the following algorithm

xn+1 PC 1 minus αn( 1113857 xn minus cnablaf xn( 1113857( 11138571113858 1113859 (16)

where c isin (0 2A2) Under some conditions it is provedthat the sequence generated by the algorithm (16) stronglyconverges to the minimum-norm solution of the (SFP)

2 Journal of Mathematics

Motivated and inspired by the work of [7 27ndash29] the au-thors of [30] introduced a self-adaptive CQ-type algorithmfor finding a solution of the (SFP) in the setting of infinite-dimensional real Hilbert spaces the advantage of this al-gorithm lies in the fact that step sizes are dynamically chosenand do not depend on the operator norm)is algorithm canbe formulated as follows

xn+1 PCn1 minus βn( 1113857 xn minus λnnablafn xn( 1113857( 11138571113858 1113859 (17)

where λn (ρnfn(xn)nablafn(xn)2) It is also proved thatthe sequence generated by the algorithm (17) stronglyconverges to the minimum-norm solution of the (SFP)

under some conditionsInspired by the works mentioned above we propose a

new relaxed CQ algorithm to solve the (SFP) in a real Hilbertspace by using inertial technology )e new step size pro-posed in this algorithm is independent of the operator normin this paper and we also establish weak convergencetheorem of the proposed algorithms under some mildconditions in [31] We add the inertial term on the basis ofthe algorithm in [30] to construct a new iterative process sothat the new algorithm strongly converges to a point in thesolution set under some conditions

)e remainder of the paper is organized as followsSome useful definitions and results are collected inSection 2 for the convergence analysis of the proposedalgorithm In Section 3 new inertial algorithms of weakand strong convergence for solving SFP are proposedfollowed by the convergence analysis In Section 4 weprovide a numerical experiment to illustrate the per-formance of the proposed algorithms Finally we end thepaper with some conclusion

2 Preliminaries

Let H be a Hilbert space and let C be a nonempty closedconvex subset in H )e strong (weak) convergence of asequence xn1113864 1113865 to x is denoted by xn⟶ x(xnx) re-spectively For any sequence xn1113864 1113865 sub H ωw(xn) denotes theweak ω minus limit set of xn1113864 1113865 that is

ωw xn( 1113857 ≔ x isin H xnjx1113882 1113883 for some subsequence nj1113966 1113967 of n

(18)

Definition 1 An operator T C⟶ H is called thefollowing

(i) Nonexpansive if

Tx minus Tyle x minus y forallx y isin C (19)

(ii) Firmly nonexpansive if

Tx minus Ty2 le x minus y

2minus (I minus T)x minus (I minus T)y

2 forallx y isin C

(20)

(iii) ]-inverse strongly monotone (]-ism) if there is ]gt 0such that

langTx minus Ty x minus yrangge ]Tx minus Ty2 forallx y isin C (21)

For every element x isin H there exists a unique nearestpoint in C denoted by PCx such that

x minus PCx

min x minus y |y isin C1113864 1113865 (22)

)en operator PC is called the metric projection from H

onto C)e projection has the following well-known properties

Lemma 1 (see [32 33]) For all x y isin H and z isin C we have

(1) langx minus PCx z minus PCxrangle 0(2) PCx minus PCyle x minus y

(3) PCx minus PCy2 le langx minus y PCx minus PCyrang

(4) PCx minus z2 le x minus z2 minus (I minus PC)x2

Lemma 2 Let H be a real Hilbert space and x y z isin Ht isin R then

(1) (1 minus t)x + ty2 (1 minus t)x2 + ty2 minus t(1 minus t)

x minus y2(2) x minus y2 y minus z2 minus x minus z2 + 2langx minus y x minus zrang

Definition 2 (see [34]) Let H be a real Hilbert space and letf H⟶ (minus infininfin) be a convex function An element v isin H

is called the subgradient of f at x isin H if

langv x minus xranglef(x) minus f(x) forallx isin H (23)

)e collection of all the subgradients of f at x is calledthe subdifferential of the function f at this point which isdenoted by zf(x) that is

zf(x) v isin H langv x minus xranglef(x) minus f(x)forallx isin H1113864 1113865

(24)

Definition 3 Let f H⟶ (minus infin +infin] be a proper function

(i) f is lower semicontinuous at x if xn⟶ x implies

f(x)le lim infn⟶infin

f xn( 1113857 (25)

(ii) f is weakly lower semicontinuous at x if xnx

implies

f(x)le lim infn⟶infin

f xn( 1113857 (26)

(iii) f is lower semicontinuous on H if it is lowersemicontinuous at every point x isin H f is weaklylower semicontinuous on H if it is weakly lowersemicontinuous at every point x isin H

(iv) f is lower semicontinuous if and only if it is weaklylower semicontinuous

Journal of Mathematics 3

Lemma 3 (see [34]) Let f H⟶ (minus infin +infin] be anα-strongly convex function gten for all x y isin H

f(y) gef(x) +langξ y minus xrang +α2

y minus x2 ξ isinzf(x) (27)

Lemma 4 (see [25]) Let tgt 0 and xlowast isin H gten the followingstatements are equivalent

(1) gte point xlowast solves the problem (SFP)(2) gte point xlowast solves the fixed-point equation

xlowast

PC xlowast

minus tAlowast

I minus PQ1113872 1113873Axlowast

1113872 1113873 (28)

(3) gte point xlowast solves the variational inequality problemwith respect to the gradient of f that is find a pointx isin C such that

langnablaf(x) y minus xrangge 0 forally isin C (29)

Lemma 5 (see [16]) Let H be a real Hilbert space and letxn1113864 1113865 be a sequence in H such that there exists a nonemptyclosed and convex subset S of H satisfying the followingconditions

(i) For all z isin S limn⟶infinxn minus z exists(ii) Any weak cluster point of xn1113864 1113865 belongs to S

gten there exists xlowast isin S such that xn1113864 1113865 converges weaklyto xlowast

Lemma 6 (see [35]) Let ϕn1113864 1113865 sub [0infin) and δn1113864 1113865 sub [0infin) betwo nonnegative real sequences satisfying the followingconditions

(1) ϕn+1 minus ϕn le θn(ϕn minus ϕnminus 1) + δn

(2) 1113936infinn1 δn ltinfin

(3) θn1113864 1113865 sub [0 θ] where θ isin [0 1)

gten ϕn1113864 1113865 is a converging sequence and1113936infinn1 [ϕn+1 minus ϕn]+ltinfin where [t]+ max t 0 for any t isin R

Lemma 7 (see [36 37]) Let an1113864 1113865infinn0 and cn1113864 1113865

infinn0 be sequences

of nonnegative real numbers such that

an+1 le 1 minus βn( 1113857an + δn + cn nge 1 (30)

where βn1113864 1113865infinn0 is a sequence in (0 1) and δn1113864 1113865

infinn0 is a real

sequence Assume 1113936infinn1 cn ltinfin gten the following results

hold

(1) If δn le βnM for some Mge 0 then an1113864 1113865infinn0 is a bounded

sequence(2) If 1113936

infinn1 βn infin and lim supn⟶infin(δnβn)le 0 then

limn⟶infinan 0

Lemma 8 (see [38]) Assume that sn1113864 1113865 is a sequence ofnonnegative real numbers such that

sn+1 le 1 minus αn( 1113857sn + αnδn nge 1

sn+1 le sn minus λn + cn nge 1(31)

where αn1113864 1113865 is a sequence in (0 1) λn1113864 1113865 is a sequence ofnonnegative real numbers and δn1113864 1113865 and cn1113864 1113865 are two se-quences in R such that

(1) 1113936infinn1 αn infin

(2) limn⟶infincn 0(3) limk⟶infinλnk

0 implies lim supk⟶infinδnkle 0 for any

subsequence nk1113864 1113865 of n

gten limn⟶infinsn 0

3 Convergence Analysis

In this section we consider the (SFP) in which C is given by

C x isin H1|c(x)le 01113864 1113865 (32)

where c H1⟶ (minus infin +infin] is an α-strongly convex func-tion the set Q is given by

Q y isin H2|q(y)le 01113864 1113865 (33)

where q H2⟶ (minus infin +infin] is a β-strongly convex func-tion We assume that the solution set S of the (SFP) isnonempty and c and q are lower semicontinuous convexfunctions furthermore we also assume that zc and zq arebounded operators (ie bounded on bounded sets)

We agree to build the following sets in our algorithmsaccording to [39] that is given the n-th iterative point wnwe construct Cn as

Cn x isin H1|c wn( 1113857 +langξn x minus wnrang +α2

x minus wn

2 le 01113882 1113883

(34)

where ξn isinzc(wn)

Qn y isin H2|q Awn( 1113857 +langζn y minus Awnrang +β2

y minus Awn

2 le 01113896 1113897

(35)

where ζn isinzq(Awn)If α 0 and β 0 then Cn and Qn are reduced to the

half-spaces Cn and Qn respectively If αgt 0 and βgt 0 thenCn and Qn are nonempty closed ball of radius

(1α)

ξn2 minus 2αc(wn)

1113969

centred at wn minus (1α)ξn and

(1β)

ζn2 minus 2βq(Awn)

1113969

centred at Awn minus (1β)ζnrespectively

In addition for each nge 0 we define the followingfunctions

fn(x) 12

I minus PQn1113872 1113873Ax

2

nablafn(x) Alowast

I minus PQn1113872 1113873Ax

(36)

where Qn is given as in (35) fn is weakly lower semi-continuous convex and differentiable and its gradient nablafn

4 Journal of Mathematics

is Lipschitz continuous Now we propose new relaxed CQ

algorithms for solving the (SFP)Next two inertial relaxed CQ algorithms will be in-

troduced )e weak convergence of Algorithm 1 and thestrong convergence of Algorithm 2 will be proved underdifferent step sizes

Algorithm 1 Choose positive sequence εn1113864 1113865 satisfying1113936infinn0 εn ltinfinLet x0 x1 isin C be arbitrary Given xn xnminus 1 update the

next iteration via

wn xn + θn xn minus xnminus 1( 1113857

xn+1 PCnwn minus τnnablafn wn( 1113857( 1113857

(37)

where 0le θn lt θn and

θn

min θεn

max xn minus xnminus 1

2 xn minus xnminus 1

1113882 1113883

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ if xn ne xnminus 1

θ if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(38)

and Cn and Qn are given as in (34) and (35)

τn

σn

nablafn wn( 1113857

if nablafn wn( 1113857

ne 0

0 if nablafn wn( 1113857

0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(39)

where 1113936infinn1 σn infin 1113936

infinn1 σ

2n ltinfin

If xn+1 wn then stop otherwise set n n + 1 and goto the next iteration

By assuming θn we know

1113944

infin

n1θn xn minus xnminus 1

2 ltinfin

1113944

infin

n1θn xn minus xnminus 1

ltinfin

(40)

which means

limn⟶infin

θn xn minus xnminus 1

2

0

limn⟶infin

θn xn minus xnminus 1

0(41)

From ξn isinzc(wn) applying Lemma 3 we get CsubeCn anda similar way is used to get QsubeQn

Now let us show that our proposed algorithm has a veryimportant property if xn+1 wn for some ngt 0 then wn is asolution of (SFP) Indeed xn+1 isin Cn so that wn isin Cn as wn

xn+1 by assumption So we get c(wn)le 0 from (34) that iswn isin C On the other hand according to the algorithm wehave wn PCn

(wn minus τnAlowast(I minus PQn)Awn) which together

with Lemma 4 implies that Awn isin Qn It also implies thatq(Awn)le 0 from (35) then Awn isin Q )e conclusion istenable

Lemma 9 Let xn1113864 1113865 and wn1113864 1113865 be the sequences generated byAlgorithm 1 gten for any z isin S it follows that

xn+1 minus z

2 le wn minus z

2

+ σ2n minus4σnfn wn( 1113857

nablafn wn( 1113857

(42)

Proof For z isin S we have z isin C Az isin Q and we havez PCz PCn

z Az PQAz PQnAz

It follows from Lemma 1 that

xn+1 minus z

2

PCnwn minus τnnablafn wn( 1113857( 1113857 minus z

2

le wn minus z( 1113857 minus τnnablafn wn( 1113857

2

minus wn minus xn+1( 1113857 minus τnnablafn wn( 1113857

2

wn minus z

2

minus wn minus xn+1

2

minus 2τnlangnablafn wn( 1113857 wn minus zrang + 2τnlangnablafn wn( 1113857 wn minus xn+1rang

(43)

where

2τnlangnablafn wn( 1113857 wn minus zrang 2τnlang I minus PQn1113872 1113873Awn minus I minus PQn

1113872 1113873Az Awn minus Azrang

ge 2τn I minus PQn1113872 1113873Awn

2

4τnfn wn( 1113857

2τnlangnablafn wn( 1113857 wn minus xn+1rang le 2τn nablafn wn( 1113857

middot wn minus xn+1

le wn minus xn+1

2

+ τ2n nablafn wn( 1113857

2

(44)

Journal of Mathematics 5

Hence we have

xn+1 minus z

2 le wn minus z

2

minus wn minus xn+1

2

minus 4τnfn wn( 1113857

+ wn minus xn+1

2

+ τ2n nablafn wn( 1113857

2

(45)

If nablafn(wn) 0 then τn 0 so that

xn+1 minus z

2 le wn minus z

2 (46)

If nablafn(wn)ne 0 we have

xn+1 minus z

2 le wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 4τnfn wn( 1113857

wn minus z

2

+ σ2n minus4σnfn wn( 1113857

nablafn wn( 1113857

(47)

)e proof is complete

Theorem 1 Assume that θn satisfies the assumption gten

there exists a subsequent xnj1113882 1113883 of xn1113864 1113865 generated by Algo-

rithm 1 which weakly converges to a solution of (SFP)

Proof We first show that for any z isin S the limit ofxn minus z1113864 1113865 exists By applying Lemma 9 we have

xn+1 minus z

2 le wn minus z

2

+ σ2n minus4σnfn wn( 1113857

nablafn wn( 1113857

(48)

From the construction of wn and Lemma 2 we have

wn minus z

2

1 + θn( 1113857 xn minus z( 1113857 minus θn xnminus 1 minus z( 1113857

2

1 + θn( 1113857 xn minus z

2

minus θn xnminus 1 minus z

2

+ θn 1 + θn( 1113857 xn minus xnminus 1

2

(49)

le 1 + θn( 1113857 xn minus z

2

minus θn xnminus 1 minus z

2

+ 2θn xn minus xnminus 1

2

(50)

Combining (48) and (50) immediately we get

xn+1 minus z

2 le 1 + θn( 1113857 xn minus z

2

minus θn xnminus 1 minus z

2

+ 2θn xn minus xnminus 1

2

+ σ2n(51)

Denote ϕn xn minus z2 from (51) we have

ϕn+1 minus ϕn le θn ϕn minus ϕnminus 1( 1113857 + 2θn xn minus xnminus 1

2

+ σ2n (52)

where

1113944

infin

n1θn xn minus xnminus 1

2 ltinfin

1113944

infin

n1σ2n ltinfin

(53)

Using Lemma 6 the limit of ϕn exists and1113936infinn1 (xn+1 minus z2 minus xn minus z2)+ltinfin which implies that

1113936infinn1(xn+1 minus z2 minus xn minus z2)ltinfin (xn+1 minus z2 minus xn

minus z2)+ max xn+1 minus z2 minus xn minus z2 01113966 1113967 )is also impliesthat the sequence xn1113864 1113865 is bounded so wn1113864 1113865 is bounded

We next show that ωw(xn) sub S Since wn1113864 1113865 is boundedfrom the Lipschitz continuity of nablafn we get thatnablafn(wn)1113864 1113865 is bounded From (48) and (50) we get

4σnfn wn( 1113857

nablafn wn( 1113857

le xn minus z

2

minus xn+1 minus z

2

+ θn xn minus z

2

minus xnminus 1 minus z

2

1113874 1113875 + 2θn xn minus xnminus 1

2

+ σ2n (54)

where 1113936infinn1(xn+1 minus z2 minus xn minus z2)ltinfin 1113936

infinn1 θnxnminus

xnminus 12 ltinfin and 1113936

infinn1 σ2n ltinfin so we have

1113944

infin

n1

4σnfn wn( 1113857

nablafn wn( 1113857

ltinfin (55)

But 1113936infinn1 σn infin so

lim infn⟶infin

fn wn( 1113857 0

ie lim infn⟶infin

I minus PQn1113872 1113873Awn

2

0(56)

On the other hand since xn1113864 1113865 is bounded the set ωw(xn)

is nonempty Let xlowast isin ωw(xn) then there exists a subse-quence xnk

1113966 1113967 of xn1113864 1113865 such that xnkxlowast Furthermore

wn minus xn

2

θ2n xn minus xnminus 1

2 le θn xn minus xnminus 1

2⟶ 0

(57)

Let wnj1113882 1113883 be a subsequence of the sequence wn1113864 1113865 such

that

lim infn⟶infin

I minus PQn1113872 1113873Awn

2

limj⟶infin

I minus PQnj1113874 1113875Awnj

2 0

(58)

Since wnj1113882 1113883 is bounded there exists a subsequence wnjm

1113882 1113883

of wnj1113882 1113883 which converges weakly to xlowast Without loss of

generality we can assume that wnjxlowast and A is a bounded

linear operator so AwnjAxlowast

From Lemma 1 we conclude that

langwn minus τnnablafn wn( 1113857 minus xn+1 z minus xn+1rang le 0 (59)

Since τn⟶ 0 and nablafn(wn)1113864 1113865 is bounded we haveτnnablafn(wn)⟶ 0 Hence we get

6 Journal of Mathematics

langxn+1 minus wn xn+1 minus zrang le langτnnablafn wn( 1113857 z minus xn+1rang⟶ 0

(60)

Since 1113936infinn1(xn+1 minus z2 minus xn minus z2)ltinfin and

1113936infinn1 θnxn minus xnminus 1

2 ltinfin from (50) we obtain

xn+1 minus wn

2

wn minus z

2

minus xn+1 minus z

2

+ 2langxn+1 minus wn xn+1 minus zrang

le xn minus z

2

minus xn+1 minus z

2

+ θn xn minus z

2

minus xnminus 1 minus z

2

1113874 1113875 + 2θn xn minus xnminus 1

2

+ 2langxn+1 minus wn xn+1 minus zrang⟶ 0

(61)

)us

xn+1 minus xn

le xn+1 minus wn

+ wn minus xn

⟶ 0 (62)

Since PQnjAwnjisin Qnj

by the definition of Qnj

q Awnj1113874 1113875 +langζnj

PQnj

Awnjminus Awnjrang +

β2

PQnj

Awnjminus Awnj

2le 0

(63)

where ζnjisinzq(Awnj

) From the boundedness assumption ofzq and limj⟶infin(I minus PQnj

)Awnj2 0 we have

q Awnj1113874 1113875le ζnj

middot I minus PQnj1113874 1113875Awnj

minusβ2

I minus PQnj1113874 1113875Awnj

2⟶ 0

(64)

From the weak lower semicontinuity of the convexfunction q it follows that

q Axlowast

( 1113857le lim infj⟶infin

q Awnj1113874 1113875le 0 (65)

which means that Axlowast isin QFurthermore xnj+1 isin Cnj

and by the definition of Cnj

c wnj1113874 1113875 +langξnj

xnj+1 minus wnjrang +

α2

xnj+1 minus wnj

2le 0 (66)

where ξnjisinzc(wnj

) From the boundedness assumption of zc

and xnj+1 minus wnj⟶ 0 we have

c wnj1113874 1113875le ξnj

middot wnjminus xnj+1

minusα2

xnj+1 minus wnj

2⟶ 0

(67)

From the weak lower semicontinuity of the convexfunction c it follows that

c xlowast

( 1113857le lim infj⟶infin

c wnj1113874 1113875le 0 (68)

which means that xlowast isin C )erefore xnjxlowast isin S )e proof

is complete

Algorithm 2 Choose positive sequence εn1113864 1113865 satisfying1113936infinn0 εn ltinfinLet x0 x1 isin C be arbitrary Given xn xnminus 1 update the

next iteration via

wn xn + θn xn minus xnminus 1( 1113857

xn+1 βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859

τn ρnfn wn( 1113857

nablafn wn( 1113857

2

(69)

where 0le θn lt θn and

θn

min θεn

max xn minus xnminus 1

2 xn minus xnminus 1

1113882 1113883

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ if xn nexnminus 1

θ if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(70)

and Cn and Qn are given as in (34) and (35) βn1113864 1113865 sub (0 1)limn⟶infinβn 0 1113936

infinn1 βn infin and infnρn(4 minus ρn)gt 0

If xn+1 wn then stop otherwise set n n + 1 and goto the next iteration

Theorem 2 Assume that infnρn(4 minus ρn)gt 0 and εn o(βn)gten the sequence xn generated by Algorithm 2 convergesstrongly to z PSu

Proof First we show that for any z isin S the sequence xn1113864 1113865

is bounded From the construction of wn we have

Journal of Mathematics 7

wn minus z

xn + θn xn minus xnminus 1( 1113857 minus z

le xn minus z

+ θn xn minus xnminus 1

(71)

wn minus τnnablafn wn( 1113857 minus z

2

wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 2τnlangnablafn wn( 1113857 wn minus zrang

le wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 4τnfn wn( 1113857

wn minus z

2

+ρ2nf

2n wn( 1113857

nablafn wn( 1113857

2 minus

4ρnf2n wn( 1113857

nablafn wn( 1113857

2

wn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

le wn minus z

2

(72)

So combining (71) and (72) we get

xn+1 minus z

βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus z

PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z + βn(u minus z)

le PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z

+ βnu minus z

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 1113857 minus 1 minus βn( 1113857z

+ βnu minus z

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus z

+ βnu minus z

le 1 minus βn( 1113857 wn minus z

+ βnu minus z

le 1 minus βn( 1113857 xn minus z

+ θn xn minus xnminus 1

1113960 1113961 + βnu minus z

le 1 minus βn( 1113857 xn minus z

+ βn σn +u minus z1113858 1113859

(73)

where σn (1 minus βn)(θnβn)xn minus xnminus 1 According to hy-pothesis θn

θn leεn

xn minus xnminus 1

rArrθn

βn

xn minus xnminus 1

leεn

βn

⟶ 0 (74)

Note that

limn⟶infin

σn limn⟶infin

1 minus βn( 1113857θn

βn

xn minus xnminus 1

0 (75)

which implies that the sequence σn1113864 1113865 is bounded Setting

M max supnisinN

σn u minus z1113896 1113897 (76)

as well as using Lemma 7 we conclude that the sequencexn minus z1113864 1113865 is bounded )is shows that the sequence xn1113864 1113865 isbounded and so is wn1113864 1113865

Since xn minus z1113864 1113865 is bounded assume that there exists aconstant M1 such that xn minus zleM1 )us

wn minus z

2 le xn minus z

+ θn xn minus xnminus 1

1113872 1113873

2

xn minus z

2

+ θ2n xn minus xnminus 1

2

+ 2θn xn minus xnminus 1

middot xn minus z

le xn minus z

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

(77)

and we get

8 Journal of Mathematics

xn+1 minus z

2

βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus z

2

PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z + βn(u minus z)

2

le PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z

2

+ 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 1113857 minus 1 minus βn( 1113857z

2

+ 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 wn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

⎡⎢⎢⎣ ⎤⎥⎥⎦ + 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 xn minus z

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

⎡⎢⎢⎣ ⎤⎥⎥⎦ + 2βnlangu minus z xn+1 minus zrang

(78)

From (78)

xn+1 minus z

2 le 1 minus βn( 1113857 xn minus z

2

+ βn

θn

βn

xn minus xnminus 1

2

+ 2M11113890

middotθn

βn

xn minus xnminus 1

+ 2langu minus z xn+1 minus zrang1113891

xn+1 minus z

2 le xn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang(79)

Let

sn xn minus z

2

δn θn

βn

xn minus xnminus 1

2

+ 2M1 middotθn

βn

xn minus xnminus 1

+ 2langu minus z xn+1 minus zrang

ηn ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

cn θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang

(80)

)en (78) can reduce to the inequalities

sn+1 le 1 minus βn( 1113857sn + βnδn nge 1

sn+1 le sn minus ηn + cn(81)

Furthermore we know that

1113944

infin

n0βn infin (82)

limn⟶infin

cn limn⟶infin

θn xn minus xnminus 1

2

+ 2M11113876

middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang1113961 0

(83)

Let nk1113864 1113865 be a subsequence of n and suppose that

limk⟶infin

ηnk 0 (84)

)en we have

limk⟶infin

ρnk4 minus ρnk

1113872 1113873f2nk

wnk1113872 1113873

nablafnkwnk

1113872 1113873

2 0 (85)

which implies by our assumption that

f2nk

wnk1113872 1113873

nablafnkwnk

1113872 1113873

2⟶ 0 as k⟶infin (86)

Since nablafnk(wnk

)1113966 1113967 is bounded it follows thatfnk

(wnk)⟶ 0 as k⟶infin so we get

limk⟶infin(I minus PQnk

)Awnk 0

We next show that ωw(xn) sub S Since xn1113864 1113865 is boundedthe set ωw(xn) is nonempty Let xlowast isin ωw(xn) then thereexists a subsequence xnk

1113966 1113967 of xn1113864 1113865 such that xnkxlowast

wn minus xn

xn + θn xn minus xnminus 1( 1113857 minus xn

θn xn minus xnminus 1

⟶ 0

(87)

and then wnjxlowast and A is a bounded linear operator so

AwnjAxlowast

Since PQnjAwnjisin Qnj

we have

q Awnj1113874 1113875 +langζnj

PQnj

Awnjminus Awnjrang +

β2

PQnj

Awnjminus Awnj

2le 0

(88)

Journal of Mathematics 9

where ζnjisinzq(Awnj

) and by the boundedness of zq we get

q Awnj1113874 1113875le ζnj

middot I minus PQnj1113874 1113875Awnj

minusβ2

I minus PQnj1113874 1113875Awnj

2⟶ 0

(89)

and using the weak lower semicontinuity of q

q Axlowast

( 1113857le lim infj⟶infin

q Awnj1113874 1113875le 0 (90)

)us Axlowast isin QOn the other hand

xn+1 minus wn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus wn

+ βnu minus z

1 minus βn( 1113857 middotρnfn wn( 1113857

nablafn wn( 1113857

+ βnu minus z⟶ 0

(91)

Since (xnj+1 minus βnju) isin Cnj

we have

c wnj1113874 1113875 +langξnj

xnj+1 minus βnju minus wnjrang +

α2

xnj+1 minus βnju minus wnj

2le 0

(92)

where ξnjisinzc(wnj

) and by the boundedness of zc we get

c wnj1113874 1113875le ξnj

middot wnjminus xnj+1 + βnj

u

minusα2

xnj+1 minus wnjminus βnj

u

2

le ξnj

middot wnjminus xnj+1

+ βnju1113876 1113877

minusα2

xnj+1 minus wnj

2

+ β2nju

2minus 2langxnj+1 minus wnj

βnjurang1113876 1113877⟶ 0

(93)

and using the weak lower semicontinuity of c

c xlowast

( 1113857le lim infj⟶infin

c wnj1113874 1113875le 0 (94)

)us xlowast isin C then xlowast isin S that is ωw(xn) sub SNext we have

xn+1 minus xn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus xn

+ βn u minus xn

le 1 minus βn( 1113857 wn minus xn

+ τn nablafn wn( 1113857

1113960 1113961 + βn xn minus u

le wn minus xn

+

ρnfn wn( 1113857

nablafn wn( 1113857

+ βn xn minus u

⟶ 0

(95)

For z PSu and xnkxlowast isin S using Lemma 1

langu minus z xlowast minus zrangle 0 so

lim supn⟶infinlangu minus z xn minus zrang lim sup

k⟶infinlangu minus z xnk

minus zrang

langu minus z xlowast

minus zrang le 0

(96)

and then

lim supn⟶infinlangu minus z xn+1 minus zrang

lim supn⟶infinlangu minus z xn+1 minus xnrang +langu minus z xn minus zrang( 1113857le 0

(97)

and thus

lim supk

δnk lim sup

k

θnk

βnk

xnkminus xnkminus 1

2

+ 2M11113890

middotθnk

βnk

xnkminus xnkminus 1

+ 2langu minus z xnk+1 minus zrang1113891le 0

(98)

From (82) (83) (98) and Lemma 8 we conclude that thesequence xn1113864 1113865 converges strongly to z PSu )e proof iscomplete

4 Numerical Experiments

In this section we present a numerical experiment to il-lustrate the performance of the proposed algorithms Ournumerical experiments are coded in MATLAB R2007running on personal computer with 350GHz Intel Core i3and 4GB RAM In what follows we apply our algorithms tosolve the problem of least absolute shrinkage and selectionoperator which requires solving a convex optimizationproblem as

minxisinRn

12Ax minus y

2

st x1 le t0

(99)

where A isin Rmtimesn y isin Rm and t0 gt 0 are given elements Inour experiment we first generate an m times n matrix A ran-domly by a standardized normal distribution and x is asparse signal with n elements only K of which is nonzerowhich is also generated randomly )e observation y isgenerated as y Ax )e parameters in this experiment areset with n 512 m 256 ε 10minus 4 and t0 K In thissituation it is readily seen that C x isin Rn c(x)le 0 withc(x) x1 minus t0 and Q y1113864 1113865 which in turn implies that

Cn x isin Rn langξn xrang le langξn wnrang minus wn

1 + t01113966 1113967 (100)

where ξn is defined by

ξn( 1113857i

1 if ξn( 1113857igt 0

[minus 1 1] if ξn( 1113857i 0

minus 1 if ξn( 1113857ilt 0

⎧⎪⎪⎨

⎪⎪⎩(101)

standing for the subdifferential of middot 1 As a half-space theassociated projection onto Cn takes the following form

10 Journal of Mathematics

PCn(x)

x +langξn wn minus xrang minus wn

1 + t0

ξn

2 ξn if x notin Cn

x if x isin Cn

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(102)

To show the efficiency of our algorithm we compareit with the algorithm proposed in [40] )e only dif-ference of these two algorithms is that there are no in-ertial terms in the algorithm proposed in [40] For theconvenience we denote Algorithm 1 by Algo I and thealgorithm in [40] by Algo II respectively In Algo-rithm 1 we set

θn

min 081

n2

xn minus xnminus 1

2

⎧⎨

⎩

⎫⎬

⎭ if xn ne xnminus 1

08 if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

τn

1n Alowast

Awn minus y( 1113857

if A

lowastAwn minus y( 1113857

ne 0

0 if Alowast

Awn minus y( 1113857

0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(103)

In Algo II we set θn equiv 0 and τn is chosen the same asabove )e stopping criterion is that xk+1 minus xklt ε )einitial points are x0 (0 0 0)T andx1 100(1 1 1)T )e numerical results of these twoalgorithms with different choices of the sparsity number K

are listed in Figures 1ndash4 It is easy to see that Algo Iconverges faster than Algo II does which indicates that ourmodified algorithm indeed accelerates the convergence ofthe original algorithm

106

104

102

100

||xR

ndash x R

+1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 1 Iterative results with K 50

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 2 Iterative results with K 40

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 3 Iterative results with K 30

104

102

100

||xR

ndash x R

ndash1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 4 Iterative results with K 20

Journal of Mathematics 11

5 Conclusions

In this paper we present two inertial relaxed CQ algorithmsfor solving split feasibility problems in Hilbert spaces byadopting variable step size )ese algorithms adopt the newconvex subset form and it is easy to calculate the projectionsonto these sets Furthermore step size selection in the al-gorithms does not depend on the operator norm )econvergence theorems are established under some mildconditions and a numerical experiment is given to illustratethe performance of the proposed algorithms

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was partially supported by the National NaturalScience Foundation of China (no 11771126)

References

[1] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquo Numerical Algo-rithms vol 8 no 2 pp 221ndash239 1994

[2] C Byrne ldquoA unified treatment of some iterative algorithms insignal processing and image reconstructionrdquo Inverse Prob-lems vol 20 no 1 pp 103ndash120 2004

[3] Y Censor T Elfving N Kopf and T Bortfeld ldquo)emultiple-sets split feasibility problem and its applications for inverseproblemsrdquo Inverse Problems vol 21 no 6 pp 2071ndash20842005

[4] C Byrne ldquoIterative oblique projection onto convex sets andthe split feasibility problemrdquo Inverse Problems vol 18 no 2pp 441ndash453 2002

[5] S He H Tian and H K Xu ldquo)e selective projection methodfor convex feasibility and split feasibility problemsrdquo Journal ofNonlinear and Convex Analysis vol 19 pp 1199ndash1215 2018

[6] S He Z Zhao and B Luo ldquoA relaxed self-adaptive CQ al-gorithm for the multiple-sets split feasibility problemrdquo Op-timization vol 64 no 9 pp 1907ndash1918 2015

[7] G L Acedo V Martin-Marquez F Wang and H XuldquoSolving the split feasibility problem without prior knowledgeof matrix normsrdquo Inverse Problems vol 28 Article ID 0850042012

[8] B Qu and N Xiu ldquoA new halfspace-relaxation projectionmethod for the split feasibility problemrdquo Linear Algebra andits Applications vol 428 no 5-6 pp 1218ndash1229 2008

[9] F Wang ldquoA splitting-relaxed projection method for solvingthe split feasibility problemrdquo Fixed Point gteory vol 14pp 211ndash218 2013

[10] F Wang ldquoOn the convergence of CQ algorithm with variablesteps for the split equality problemrdquo Numerical Algorithmsvol 74 no 3 pp 927ndash935 2017

[11] F Wang ldquoPolyakrsquos gradient method for split feasibilityproblem constrained by level setsrdquo Numerical Algorithmsvol 77 no 3 pp 925ndash938 2018

[12] H-K Xu ldquoA variable Krasnoselrsquoskii-Mann algorithm and themultiple-set split feasibility problemrdquo Inverse Problemsvol 22 no 6 pp 2021ndash2034 2006

[13] Q Yang ldquo)e relaxed CQ algorithm solving the split feasi-bility problemrdquo Inverse Problems vol 20 no 4pp 1261ndash1266 2004

[14] F Alvarez and H Attouch ldquoAn inertial proximal method formaximal monotone operators via discretization of a nonlinearoscillator with dampingrdquo Set-Valued Analysis vol 9 pp 3ndash112001

[15] Y Dang J Sun J Sun and H Xu ldquoInertial accelerated al-gorithms for solving a split feasibility problemrdquo Journal ofIndustrial amp Management Optimization vol 13 no 3pp 1383ndash1394 2017

[16] A Gibali D Mai D )i Mai and N )e Vinh ldquoA newrelaxed CQ algorithm for solving split feasibility problems inHilbert spaces and its applicationsrdquo Journal of Industrial ampManagement Optimization vol 15 no 2 pp 963ndash984 2019

[17] Q L Dong Y J Cho and T M Rassias ldquoGeneral inertialMann algorithms and their convergence analysis for non-expansive mappingsrdquo in Applications of Nonlinear Analysispp 175ndash191 Springer Berlin Germany 2018

[18] Q L Dong Y J Cho L L Zhong and T M Rassias ldquoInertialprojection and contraction algorithms for variational in-equalitiesrdquo Journal of Global Optimization vol 70 no 3pp 687ndash704 2018

[19] Q L Dong J Z Huang X H Li Y J Cho and T M RassiasldquoMiKM multi-step inertial Krasnoselrsquoskiǐ-Mann algorithmand its applicationsrdquo Journal of Global Optimization vol 73no 4 pp 801ndash824 2019

[20] Q L Dong H B Yuan Y J Cho and T M RassiasldquoModified inertial Mann algorithm and inertial CQ-algorithmfor nonexpansive mappingsrdquo Optimization Letters vol 12no 1 pp 87ndash102 2018

[21] Y Shehu and A Gibali ldquoNew inertial relaxed method forsolving split feasibilitiesrdquo Optimization Letters 2020

[22] S S Tai N Pholasa and P Cholamjiak ldquo)emodified inertialrelaxed CQ algorithm for solving the split feasibility prob-lemsrdquo Journal of Industrial and Management Optimizationvol 14 pp 1595ndash1615 2018

[23] S S Tai N Pholasa and P Cholamjiak ldquoRelaxed CQ algo-rithms involving the inertial technique for multiple-sets splitfeasibility problemsrdquo Revista de la Real Academia de CienciasExactas Fisicas y Naturales-Serie A Matematicas vol 113pp 1081ndash1099 2019

[24] S Y Cho ldquoA convergence theorem for generalized mixedequilibrium problems and multivalued asymptotically non-expansive mappingsrdquo Journal of Nonlinear and ConvexAnalysis vol 21 pp 1017ndash1026 2020

[25] H-K Xu ldquoIterative methods for the split feasibility problemin infinite-dimensional Hilbert spacesrdquo Inverse Problemsvol 26 no 10 Article ID 105018 2010

[26] F Wang and H Xu ldquoApproximating curve and strongconvergence of the CQ algorithm for the split feasibilityproblemrdquo Journal of Inequalities and Applications vol 2010Article ID 102085 2010

[27] M Tian and H F Zhang ldquo)e regularized CQ algorithmwithout a priori knowledge of operator norm for solving thesplit feasibility problemrdquo Journal of Inequalities and Appli-cations vol 2017 p 207 2017

[28] Y H Yao W J Gang and Y C Liou ldquoRegularized methodsfor the split feasibility problemrdquo Abstract and AppliedAnalysis vol 2012 Article ID 140679 2012

12 Journal of Mathematics

[29] Y Zhou Z Haiyun and P Wang ldquoIterative methods forfinding the minimum-norm solution of the standardmonotone variational inequality problems with applicationsin Hilbert spacesrdquo Journal of Inequalities and Applicationsvol 2015 2015

[30] T V Nguyen C Prasit and S Suthep ldquoA new CQ algorithmfor solving split feasibility problems in Hilbert spacesrdquoMalaysian Mathematical Sciences Society and Penerbit Uni-versiti Sains Malaysia vol 42 no 5 2020

[31] D R Sahu Y J Cho Q L Dong M R Kashyap and X H LiInertial Relaxed CQ algorithms for Solving a Split FeasibilityProblem in Hilbert Spaces Springer Science Business MediaBerlin Germany 2020

[32] R P Agarwal D O Regan and D R Sahu ldquoFixed pointtheory for Lipschitzian-type mappings with applicationsrdquo inTopological Fixed Point gteory and Its Applications SpringerNew York NY USA 2009

[33] S Y Cho ldquoAmonotone Bregan projection algorithm for fixedpoint and equilibrium problems in a relative Banach spacerdquoFilomat vol 34 pp 1487ndash1497 2020

[34] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator gteory in Hilbert Spaces Springer NewYork NY USA 2011

[35] P E Mainge ldquoInertial iterative process for fixed points ofcertain quasi-nonexpansive mappingrdquo Set Valued Analysisvol 15 pp 67ndash79 2007

[36] P E Mainge ldquoApproximation methods for common fixedpoints of nonexpansive mappings in Hilbert spacesrdquo Journalof Mathematical Analysis and Applications vol 325pp 469ndash479 2007

[37] A Moudafi and A Gibali ldquoregularization of split feasibilityproblemsrdquo Numerical Algorithmsvol 78 no 3 pp 1ndash192017

[38] S He and C Yang ldquoSolving the variational inequality problemdefined on intersection of finite level setsrdquo Abstract andApplied Analysis vol 2013 Article ID 942315 2013

[39] F Wang and H Yu ldquoAn inertial relaxed CQ algorithm withan application to the LASSO and elastic netrdquo Optimizationvol 2020 Article ID 1763989 2020

[40] P K Anh N T Vinh and V T Dung ldquoA new self-adaptiveCQ algorithm with an application to the LASSO problemrdquoJournal of Fixed Point gteory and Applications vol 20 p 1422018

Journal of Mathematics 13

Motivated and inspired by the work of [7 27ndash29] the au-thors of [30] introduced a self-adaptive CQ-type algorithmfor finding a solution of the (SFP) in the setting of infinite-dimensional real Hilbert spaces the advantage of this al-gorithm lies in the fact that step sizes are dynamically chosenand do not depend on the operator norm)is algorithm canbe formulated as follows

xn+1 PCn1 minus βn( 1113857 xn minus λnnablafn xn( 1113857( 11138571113858 1113859 (17)

where λn (ρnfn(xn)nablafn(xn)2) It is also proved thatthe sequence generated by the algorithm (17) stronglyconverges to the minimum-norm solution of the (SFP)

under some conditionsInspired by the works mentioned above we propose a

new relaxed CQ algorithm to solve the (SFP) in a real Hilbertspace by using inertial technology )e new step size pro-posed in this algorithm is independent of the operator normin this paper and we also establish weak convergencetheorem of the proposed algorithms under some mildconditions in [31] We add the inertial term on the basis ofthe algorithm in [30] to construct a new iterative process sothat the new algorithm strongly converges to a point in thesolution set under some conditions

)e remainder of the paper is organized as followsSome useful definitions and results are collected inSection 2 for the convergence analysis of the proposedalgorithm In Section 3 new inertial algorithms of weakand strong convergence for solving SFP are proposedfollowed by the convergence analysis In Section 4 weprovide a numerical experiment to illustrate the per-formance of the proposed algorithms Finally we end thepaper with some conclusion

2 Preliminaries

Let H be a Hilbert space and let C be a nonempty closedconvex subset in H )e strong (weak) convergence of asequence xn1113864 1113865 to x is denoted by xn⟶ x(xnx) re-spectively For any sequence xn1113864 1113865 sub H ωw(xn) denotes theweak ω minus limit set of xn1113864 1113865 that is

ωw xn( 1113857 ≔ x isin H xnjx1113882 1113883 for some subsequence nj1113966 1113967 of n

(18)

Definition 1 An operator T C⟶ H is called thefollowing

(i) Nonexpansive if

Tx minus Tyle x minus y forallx y isin C (19)

(ii) Firmly nonexpansive if

Tx minus Ty2 le x minus y

2minus (I minus T)x minus (I minus T)y

2 forallx y isin C

(20)

(iii) ]-inverse strongly monotone (]-ism) if there is ]gt 0such that

langTx minus Ty x minus yrangge ]Tx minus Ty2 forallx y isin C (21)

For every element x isin H there exists a unique nearestpoint in C denoted by PCx such that

x minus PCx

min x minus y |y isin C1113864 1113865 (22)

)en operator PC is called the metric projection from H

onto C)e projection has the following well-known properties

Lemma 1 (see [32 33]) For all x y isin H and z isin C we have

(1) langx minus PCx z minus PCxrangle 0(2) PCx minus PCyle x minus y

(3) PCx minus PCy2 le langx minus y PCx minus PCyrang

(4) PCx minus z2 le x minus z2 minus (I minus PC)x2

Lemma 2 Let H be a real Hilbert space and x y z isin Ht isin R then

(1) (1 minus t)x + ty2 (1 minus t)x2 + ty2 minus t(1 minus t)

x minus y2(2) x minus y2 y minus z2 minus x minus z2 + 2langx minus y x minus zrang

Definition 2 (see [34]) Let H be a real Hilbert space and letf H⟶ (minus infininfin) be a convex function An element v isin H

is called the subgradient of f at x isin H if

langv x minus xranglef(x) minus f(x) forallx isin H (23)

)e collection of all the subgradients of f at x is calledthe subdifferential of the function f at this point which isdenoted by zf(x) that is

zf(x) v isin H langv x minus xranglef(x) minus f(x)forallx isin H1113864 1113865

(24)

Definition 3 Let f H⟶ (minus infin +infin] be a proper function

(i) f is lower semicontinuous at x if xn⟶ x implies

f(x)le lim infn⟶infin

f xn( 1113857 (25)

(ii) f is weakly lower semicontinuous at x if xnx

implies

f(x)le lim infn⟶infin

f xn( 1113857 (26)

(iii) f is lower semicontinuous on H if it is lowersemicontinuous at every point x isin H f is weaklylower semicontinuous on H if it is weakly lowersemicontinuous at every point x isin H

(iv) f is lower semicontinuous if and only if it is weaklylower semicontinuous

Journal of Mathematics 3

Lemma 3 (see [34]) Let f H⟶ (minus infin +infin] be anα-strongly convex function gten for all x y isin H

f(y) gef(x) +langξ y minus xrang +α2

y minus x2 ξ isinzf(x) (27)

Lemma 4 (see [25]) Let tgt 0 and xlowast isin H gten the followingstatements are equivalent

(1) gte point xlowast solves the problem (SFP)(2) gte point xlowast solves the fixed-point equation

xlowast

PC xlowast

minus tAlowast

I minus PQ1113872 1113873Axlowast

1113872 1113873 (28)

(3) gte point xlowast solves the variational inequality problemwith respect to the gradient of f that is find a pointx isin C such that

langnablaf(x) y minus xrangge 0 forally isin C (29)

Lemma 5 (see [16]) Let H be a real Hilbert space and letxn1113864 1113865 be a sequence in H such that there exists a nonemptyclosed and convex subset S of H satisfying the followingconditions

(i) For all z isin S limn⟶infinxn minus z exists(ii) Any weak cluster point of xn1113864 1113865 belongs to S

gten there exists xlowast isin S such that xn1113864 1113865 converges weaklyto xlowast

Lemma 6 (see [35]) Let ϕn1113864 1113865 sub [0infin) and δn1113864 1113865 sub [0infin) betwo nonnegative real sequences satisfying the followingconditions

(1) ϕn+1 minus ϕn le θn(ϕn minus ϕnminus 1) + δn

(2) 1113936infinn1 δn ltinfin

(3) θn1113864 1113865 sub [0 θ] where θ isin [0 1)

gten ϕn1113864 1113865 is a converging sequence and1113936infinn1 [ϕn+1 minus ϕn]+ltinfin where [t]+ max t 0 for any t isin R

Lemma 7 (see [36 37]) Let an1113864 1113865infinn0 and cn1113864 1113865

infinn0 be sequences

of nonnegative real numbers such that

an+1 le 1 minus βn( 1113857an + δn + cn nge 1 (30)

where βn1113864 1113865infinn0 is a sequence in (0 1) and δn1113864 1113865

infinn0 is a real

sequence Assume 1113936infinn1 cn ltinfin gten the following results

hold

(1) If δn le βnM for some Mge 0 then an1113864 1113865infinn0 is a bounded

sequence(2) If 1113936

infinn1 βn infin and lim supn⟶infin(δnβn)le 0 then

limn⟶infinan 0

Lemma 8 (see [38]) Assume that sn1113864 1113865 is a sequence ofnonnegative real numbers such that

sn+1 le 1 minus αn( 1113857sn + αnδn nge 1

sn+1 le sn minus λn + cn nge 1(31)

where αn1113864 1113865 is a sequence in (0 1) λn1113864 1113865 is a sequence ofnonnegative real numbers and δn1113864 1113865 and cn1113864 1113865 are two se-quences in R such that

(1) 1113936infinn1 αn infin

(2) limn⟶infincn 0(3) limk⟶infinλnk

0 implies lim supk⟶infinδnkle 0 for any

subsequence nk1113864 1113865 of n

gten limn⟶infinsn 0

3 Convergence Analysis

In this section we consider the (SFP) in which C is given by

C x isin H1|c(x)le 01113864 1113865 (32)

where c H1⟶ (minus infin +infin] is an α-strongly convex func-tion the set Q is given by

Q y isin H2|q(y)le 01113864 1113865 (33)

where q H2⟶ (minus infin +infin] is a β-strongly convex func-tion We assume that the solution set S of the (SFP) isnonempty and c and q are lower semicontinuous convexfunctions furthermore we also assume that zc and zq arebounded operators (ie bounded on bounded sets)

We agree to build the following sets in our algorithmsaccording to [39] that is given the n-th iterative point wnwe construct Cn as

Cn x isin H1|c wn( 1113857 +langξn x minus wnrang +α2

x minus wn

2 le 01113882 1113883

(34)

where ξn isinzc(wn)

Qn y isin H2|q Awn( 1113857 +langζn y minus Awnrang +β2

y minus Awn

2 le 01113896 1113897

(35)

where ζn isinzq(Awn)If α 0 and β 0 then Cn and Qn are reduced to the

half-spaces Cn and Qn respectively If αgt 0 and βgt 0 thenCn and Qn are nonempty closed ball of radius

(1α)

ξn2 minus 2αc(wn)

1113969

centred at wn minus (1α)ξn and

(1β)

ζn2 minus 2βq(Awn)

1113969

centred at Awn minus (1β)ζnrespectively

In addition for each nge 0 we define the followingfunctions

fn(x) 12

I minus PQn1113872 1113873Ax

2

nablafn(x) Alowast

I minus PQn1113872 1113873Ax

(36)

where Qn is given as in (35) fn is weakly lower semi-continuous convex and differentiable and its gradient nablafn

4 Journal of Mathematics

is Lipschitz continuous Now we propose new relaxed CQ

algorithms for solving the (SFP)Next two inertial relaxed CQ algorithms will be in-

troduced )e weak convergence of Algorithm 1 and thestrong convergence of Algorithm 2 will be proved underdifferent step sizes

Algorithm 1 Choose positive sequence εn1113864 1113865 satisfying1113936infinn0 εn ltinfinLet x0 x1 isin C be arbitrary Given xn xnminus 1 update the

next iteration via

wn xn + θn xn minus xnminus 1( 1113857

xn+1 PCnwn minus τnnablafn wn( 1113857( 1113857

(37)

where 0le θn lt θn and

θn

min θεn

max xn minus xnminus 1

2 xn minus xnminus 1

1113882 1113883

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ if xn ne xnminus 1

θ if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(38)

and Cn and Qn are given as in (34) and (35)

τn

σn

nablafn wn( 1113857

if nablafn wn( 1113857

ne 0

0 if nablafn wn( 1113857

0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(39)

where 1113936infinn1 σn infin 1113936

infinn1 σ

2n ltinfin

If xn+1 wn then stop otherwise set n n + 1 and goto the next iteration

By assuming θn we know

1113944

infin

n1θn xn minus xnminus 1

2 ltinfin

1113944

infin

n1θn xn minus xnminus 1

ltinfin

(40)

which means

limn⟶infin

θn xn minus xnminus 1

2

0

limn⟶infin

θn xn minus xnminus 1

0(41)

From ξn isinzc(wn) applying Lemma 3 we get CsubeCn anda similar way is used to get QsubeQn

Now let us show that our proposed algorithm has a veryimportant property if xn+1 wn for some ngt 0 then wn is asolution of (SFP) Indeed xn+1 isin Cn so that wn isin Cn as wn

xn+1 by assumption So we get c(wn)le 0 from (34) that iswn isin C On the other hand according to the algorithm wehave wn PCn

(wn minus τnAlowast(I minus PQn)Awn) which together

with Lemma 4 implies that Awn isin Qn It also implies thatq(Awn)le 0 from (35) then Awn isin Q )e conclusion istenable

Lemma 9 Let xn1113864 1113865 and wn1113864 1113865 be the sequences generated byAlgorithm 1 gten for any z isin S it follows that

xn+1 minus z

2 le wn minus z

2

+ σ2n minus4σnfn wn( 1113857

nablafn wn( 1113857

(42)

Proof For z isin S we have z isin C Az isin Q and we havez PCz PCn

z Az PQAz PQnAz

It follows from Lemma 1 that

xn+1 minus z

2

PCnwn minus τnnablafn wn( 1113857( 1113857 minus z

2

le wn minus z( 1113857 minus τnnablafn wn( 1113857

2

minus wn minus xn+1( 1113857 minus τnnablafn wn( 1113857

2

wn minus z

2

minus wn minus xn+1

2

minus 2τnlangnablafn wn( 1113857 wn minus zrang + 2τnlangnablafn wn( 1113857 wn minus xn+1rang

(43)

where

2τnlangnablafn wn( 1113857 wn minus zrang 2τnlang I minus PQn1113872 1113873Awn minus I minus PQn

1113872 1113873Az Awn minus Azrang

ge 2τn I minus PQn1113872 1113873Awn

2

4τnfn wn( 1113857

2τnlangnablafn wn( 1113857 wn minus xn+1rang le 2τn nablafn wn( 1113857

middot wn minus xn+1

le wn minus xn+1

2

+ τ2n nablafn wn( 1113857

2

(44)

Journal of Mathematics 5

Hence we have

xn+1 minus z

2 le wn minus z

2

minus wn minus xn+1

2

minus 4τnfn wn( 1113857

+ wn minus xn+1

2

+ τ2n nablafn wn( 1113857

2

(45)

If nablafn(wn) 0 then τn 0 so that

xn+1 minus z

2 le wn minus z

2 (46)

If nablafn(wn)ne 0 we have

xn+1 minus z

2 le wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 4τnfn wn( 1113857

wn minus z

2

+ σ2n minus4σnfn wn( 1113857

nablafn wn( 1113857

(47)

)e proof is complete

Theorem 1 Assume that θn satisfies the assumption gten

there exists a subsequent xnj1113882 1113883 of xn1113864 1113865 generated by Algo-

rithm 1 which weakly converges to a solution of (SFP)

Proof We first show that for any z isin S the limit ofxn minus z1113864 1113865 exists By applying Lemma 9 we have

xn+1 minus z

2 le wn minus z

2

+ σ2n minus4σnfn wn( 1113857

nablafn wn( 1113857

(48)

From the construction of wn and Lemma 2 we have

wn minus z

2

1 + θn( 1113857 xn minus z( 1113857 minus θn xnminus 1 minus z( 1113857

2

1 + θn( 1113857 xn minus z

2

minus θn xnminus 1 minus z

2

+ θn 1 + θn( 1113857 xn minus xnminus 1

2

(49)

le 1 + θn( 1113857 xn minus z

2

minus θn xnminus 1 minus z

2

+ 2θn xn minus xnminus 1

2

(50)

Combining (48) and (50) immediately we get

xn+1 minus z

2 le 1 + θn( 1113857 xn minus z

2

minus θn xnminus 1 minus z

2

+ 2θn xn minus xnminus 1

2

+ σ2n(51)

Denote ϕn xn minus z2 from (51) we have

ϕn+1 minus ϕn le θn ϕn minus ϕnminus 1( 1113857 + 2θn xn minus xnminus 1

2

+ σ2n (52)

where

1113944

infin

n1θn xn minus xnminus 1

2 ltinfin

1113944

infin

n1σ2n ltinfin

(53)

Using Lemma 6 the limit of ϕn exists and1113936infinn1 (xn+1 minus z2 minus xn minus z2)+ltinfin which implies that

1113936infinn1(xn+1 minus z2 minus xn minus z2)ltinfin (xn+1 minus z2 minus xn

minus z2)+ max xn+1 minus z2 minus xn minus z2 01113966 1113967 )is also impliesthat the sequence xn1113864 1113865 is bounded so wn1113864 1113865 is bounded

We next show that ωw(xn) sub S Since wn1113864 1113865 is boundedfrom the Lipschitz continuity of nablafn we get thatnablafn(wn)1113864 1113865 is bounded From (48) and (50) we get

4σnfn wn( 1113857

nablafn wn( 1113857

le xn minus z

2

minus xn+1 minus z

2

+ θn xn minus z

2

minus xnminus 1 minus z

2

1113874 1113875 + 2θn xn minus xnminus 1

2

+ σ2n (54)

where 1113936infinn1(xn+1 minus z2 minus xn minus z2)ltinfin 1113936

infinn1 θnxnminus

xnminus 12 ltinfin and 1113936

infinn1 σ2n ltinfin so we have

1113944

infin

n1

4σnfn wn( 1113857

nablafn wn( 1113857

ltinfin (55)

But 1113936infinn1 σn infin so

lim infn⟶infin

fn wn( 1113857 0

ie lim infn⟶infin

I minus PQn1113872 1113873Awn

2

0(56)

On the other hand since xn1113864 1113865 is bounded the set ωw(xn)

is nonempty Let xlowast isin ωw(xn) then there exists a subse-quence xnk

1113966 1113967 of xn1113864 1113865 such that xnkxlowast Furthermore

wn minus xn

2

θ2n xn minus xnminus 1

2 le θn xn minus xnminus 1

2⟶ 0

(57)

Let wnj1113882 1113883 be a subsequence of the sequence wn1113864 1113865 such

that

lim infn⟶infin

I minus PQn1113872 1113873Awn

2

limj⟶infin

I minus PQnj1113874 1113875Awnj

2 0

(58)

Since wnj1113882 1113883 is bounded there exists a subsequence wnjm

1113882 1113883

of wnj1113882 1113883 which converges weakly to xlowast Without loss of

generality we can assume that wnjxlowast and A is a bounded

linear operator so AwnjAxlowast

From Lemma 1 we conclude that

langwn minus τnnablafn wn( 1113857 minus xn+1 z minus xn+1rang le 0 (59)

Since τn⟶ 0 and nablafn(wn)1113864 1113865 is bounded we haveτnnablafn(wn)⟶ 0 Hence we get

6 Journal of Mathematics

langxn+1 minus wn xn+1 minus zrang le langτnnablafn wn( 1113857 z minus xn+1rang⟶ 0

(60)

Since 1113936infinn1(xn+1 minus z2 minus xn minus z2)ltinfin and

1113936infinn1 θnxn minus xnminus 1

2 ltinfin from (50) we obtain

xn+1 minus wn

2

wn minus z

2

minus xn+1 minus z

2

+ 2langxn+1 minus wn xn+1 minus zrang

le xn minus z

2

minus xn+1 minus z

2

+ θn xn minus z

2

minus xnminus 1 minus z

2

1113874 1113875 + 2θn xn minus xnminus 1

2

+ 2langxn+1 minus wn xn+1 minus zrang⟶ 0

(61)

)us

xn+1 minus xn

le xn+1 minus wn

+ wn minus xn

⟶ 0 (62)

Since PQnjAwnjisin Qnj

by the definition of Qnj

q Awnj1113874 1113875 +langζnj

PQnj

Awnjminus Awnjrang +

β2

PQnj

Awnjminus Awnj

2le 0

(63)

where ζnjisinzq(Awnj

) From the boundedness assumption ofzq and limj⟶infin(I minus PQnj

)Awnj2 0 we have

q Awnj1113874 1113875le ζnj

middot I minus PQnj1113874 1113875Awnj

minusβ2

I minus PQnj1113874 1113875Awnj

2⟶ 0

(64)

From the weak lower semicontinuity of the convexfunction q it follows that

q Axlowast

( 1113857le lim infj⟶infin

q Awnj1113874 1113875le 0 (65)

which means that Axlowast isin QFurthermore xnj+1 isin Cnj

and by the definition of Cnj

c wnj1113874 1113875 +langξnj

xnj+1 minus wnjrang +

α2

xnj+1 minus wnj

2le 0 (66)

where ξnjisinzc(wnj

) From the boundedness assumption of zc

and xnj+1 minus wnj⟶ 0 we have

c wnj1113874 1113875le ξnj

middot wnjminus xnj+1

minusα2

xnj+1 minus wnj

2⟶ 0

(67)

From the weak lower semicontinuity of the convexfunction c it follows that

c xlowast

( 1113857le lim infj⟶infin

c wnj1113874 1113875le 0 (68)

which means that xlowast isin C )erefore xnjxlowast isin S )e proof

is complete

Algorithm 2 Choose positive sequence εn1113864 1113865 satisfying1113936infinn0 εn ltinfinLet x0 x1 isin C be arbitrary Given xn xnminus 1 update the

next iteration via

wn xn + θn xn minus xnminus 1( 1113857

xn+1 βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859

τn ρnfn wn( 1113857

nablafn wn( 1113857

2

(69)

where 0le θn lt θn and

θn

min θεn

max xn minus xnminus 1

2 xn minus xnminus 1

1113882 1113883

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ if xn nexnminus 1

θ if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(70)

and Cn and Qn are given as in (34) and (35) βn1113864 1113865 sub (0 1)limn⟶infinβn 0 1113936

infinn1 βn infin and infnρn(4 minus ρn)gt 0

If xn+1 wn then stop otherwise set n n + 1 and goto the next iteration

Theorem 2 Assume that infnρn(4 minus ρn)gt 0 and εn o(βn)gten the sequence xn generated by Algorithm 2 convergesstrongly to z PSu

Proof First we show that for any z isin S the sequence xn1113864 1113865

is bounded From the construction of wn we have

Journal of Mathematics 7

wn minus z

xn + θn xn minus xnminus 1( 1113857 minus z

le xn minus z

+ θn xn minus xnminus 1

(71)

wn minus τnnablafn wn( 1113857 minus z

2

wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 2τnlangnablafn wn( 1113857 wn minus zrang

le wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 4τnfn wn( 1113857

wn minus z

2

+ρ2nf

2n wn( 1113857

nablafn wn( 1113857

2 minus

4ρnf2n wn( 1113857

nablafn wn( 1113857

2

wn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

le wn minus z

2

(72)

So combining (71) and (72) we get

xn+1 minus z

βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus z

PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z + βn(u minus z)

le PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z

+ βnu minus z

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 1113857 minus 1 minus βn( 1113857z

+ βnu minus z

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus z

+ βnu minus z

le 1 minus βn( 1113857 wn minus z

+ βnu minus z

le 1 minus βn( 1113857 xn minus z

+ θn xn minus xnminus 1

1113960 1113961 + βnu minus z

le 1 minus βn( 1113857 xn minus z

+ βn σn +u minus z1113858 1113859

(73)

where σn (1 minus βn)(θnβn)xn minus xnminus 1 According to hy-pothesis θn

θn leεn

xn minus xnminus 1

rArrθn

βn

xn minus xnminus 1

leεn

βn

⟶ 0 (74)

Note that

limn⟶infin

σn limn⟶infin

1 minus βn( 1113857θn

βn

xn minus xnminus 1

0 (75)

which implies that the sequence σn1113864 1113865 is bounded Setting

M max supnisinN

σn u minus z1113896 1113897 (76)

as well as using Lemma 7 we conclude that the sequencexn minus z1113864 1113865 is bounded )is shows that the sequence xn1113864 1113865 isbounded and so is wn1113864 1113865

Since xn minus z1113864 1113865 is bounded assume that there exists aconstant M1 such that xn minus zleM1 )us

wn minus z

2 le xn minus z

+ θn xn minus xnminus 1

1113872 1113873

2

xn minus z

2

+ θ2n xn minus xnminus 1

2

+ 2θn xn minus xnminus 1

middot xn minus z

le xn minus z

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

(77)

and we get

8 Journal of Mathematics

xn+1 minus z

2

βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus z

2

PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z + βn(u minus z)

2

le PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z

2

+ 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 1113857 minus 1 minus βn( 1113857z

2

+ 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 wn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

⎡⎢⎢⎣ ⎤⎥⎥⎦ + 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 xn minus z

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

⎡⎢⎢⎣ ⎤⎥⎥⎦ + 2βnlangu minus z xn+1 minus zrang

(78)

From (78)

xn+1 minus z

2 le 1 minus βn( 1113857 xn minus z

2

+ βn

θn

βn

xn minus xnminus 1

2

+ 2M11113890

middotθn

βn

xn minus xnminus 1

+ 2langu minus z xn+1 minus zrang1113891

xn+1 minus z

2 le xn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang(79)

Let

sn xn minus z

2

δn θn

βn

xn minus xnminus 1

2

+ 2M1 middotθn

βn

xn minus xnminus 1

+ 2langu minus z xn+1 minus zrang

ηn ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

cn θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang

(80)

)en (78) can reduce to the inequalities

sn+1 le 1 minus βn( 1113857sn + βnδn nge 1

sn+1 le sn minus ηn + cn(81)

Furthermore we know that

1113944

infin

n0βn infin (82)

limn⟶infin

cn limn⟶infin

θn xn minus xnminus 1

2

+ 2M11113876

middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang1113961 0

(83)

Let nk1113864 1113865 be a subsequence of n and suppose that

limk⟶infin

ηnk 0 (84)

)en we have

limk⟶infin

ρnk4 minus ρnk

1113872 1113873f2nk

wnk1113872 1113873

nablafnkwnk

1113872 1113873

2 0 (85)

which implies by our assumption that

f2nk

wnk1113872 1113873

nablafnkwnk

1113872 1113873

2⟶ 0 as k⟶infin (86)

Since nablafnk(wnk

)1113966 1113967 is bounded it follows thatfnk

(wnk)⟶ 0 as k⟶infin so we get

limk⟶infin(I minus PQnk

)Awnk 0

We next show that ωw(xn) sub S Since xn1113864 1113865 is boundedthe set ωw(xn) is nonempty Let xlowast isin ωw(xn) then thereexists a subsequence xnk

1113966 1113967 of xn1113864 1113865 such that xnkxlowast

wn minus xn

xn + θn xn minus xnminus 1( 1113857 minus xn

θn xn minus xnminus 1

⟶ 0

(87)

and then wnjxlowast and A is a bounded linear operator so

AwnjAxlowast

Since PQnjAwnjisin Qnj

we have

q Awnj1113874 1113875 +langζnj

PQnj

Awnjminus Awnjrang +

β2

PQnj

Awnjminus Awnj

2le 0

(88)

Journal of Mathematics 9

where ζnjisinzq(Awnj

) and by the boundedness of zq we get

q Awnj1113874 1113875le ζnj

middot I minus PQnj1113874 1113875Awnj

minusβ2

I minus PQnj1113874 1113875Awnj

2⟶ 0

(89)

and using the weak lower semicontinuity of q

q Axlowast

( 1113857le lim infj⟶infin

q Awnj1113874 1113875le 0 (90)

)us Axlowast isin QOn the other hand

xn+1 minus wn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus wn

+ βnu minus z

1 minus βn( 1113857 middotρnfn wn( 1113857

nablafn wn( 1113857

+ βnu minus z⟶ 0

(91)

Since (xnj+1 minus βnju) isin Cnj

we have

c wnj1113874 1113875 +langξnj

xnj+1 minus βnju minus wnjrang +

α2

xnj+1 minus βnju minus wnj

2le 0

(92)

where ξnjisinzc(wnj

) and by the boundedness of zc we get

c wnj1113874 1113875le ξnj

middot wnjminus xnj+1 + βnj

u

minusα2

xnj+1 minus wnjminus βnj

u

2

le ξnj

middot wnjminus xnj+1

+ βnju1113876 1113877

minusα2

xnj+1 minus wnj

2

+ β2nju

2minus 2langxnj+1 minus wnj

βnjurang1113876 1113877⟶ 0

(93)

and using the weak lower semicontinuity of c

c xlowast

( 1113857le lim infj⟶infin

c wnj1113874 1113875le 0 (94)

)us xlowast isin C then xlowast isin S that is ωw(xn) sub SNext we have

xn+1 minus xn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus xn

+ βn u minus xn

le 1 minus βn( 1113857 wn minus xn

+ τn nablafn wn( 1113857

1113960 1113961 + βn xn minus u

le wn minus xn

+

ρnfn wn( 1113857

nablafn wn( 1113857

+ βn xn minus u

⟶ 0

(95)

For z PSu and xnkxlowast isin S using Lemma 1

langu minus z xlowast minus zrangle 0 so

lim supn⟶infinlangu minus z xn minus zrang lim sup

k⟶infinlangu minus z xnk

minus zrang

langu minus z xlowast

minus zrang le 0

(96)

and then

lim supn⟶infinlangu minus z xn+1 minus zrang

lim supn⟶infinlangu minus z xn+1 minus xnrang +langu minus z xn minus zrang( 1113857le 0

(97)

and thus

lim supk

δnk lim sup

k

θnk

βnk

xnkminus xnkminus 1

2

+ 2M11113890

middotθnk

βnk

xnkminus xnkminus 1

+ 2langu minus z xnk+1 minus zrang1113891le 0

(98)

From (82) (83) (98) and Lemma 8 we conclude that thesequence xn1113864 1113865 converges strongly to z PSu )e proof iscomplete

4 Numerical Experiments

In this section we present a numerical experiment to il-lustrate the performance of the proposed algorithms Ournumerical experiments are coded in MATLAB R2007running on personal computer with 350GHz Intel Core i3and 4GB RAM In what follows we apply our algorithms tosolve the problem of least absolute shrinkage and selectionoperator which requires solving a convex optimizationproblem as

minxisinRn

12Ax minus y

2

st x1 le t0

(99)

where A isin Rmtimesn y isin Rm and t0 gt 0 are given elements Inour experiment we first generate an m times n matrix A ran-domly by a standardized normal distribution and x is asparse signal with n elements only K of which is nonzerowhich is also generated randomly )e observation y isgenerated as y Ax )e parameters in this experiment areset with n 512 m 256 ε 10minus 4 and t0 K In thissituation it is readily seen that C x isin Rn c(x)le 0 withc(x) x1 minus t0 and Q y1113864 1113865 which in turn implies that

Cn x isin Rn langξn xrang le langξn wnrang minus wn

1 + t01113966 1113967 (100)

where ξn is defined by

ξn( 1113857i

1 if ξn( 1113857igt 0

[minus 1 1] if ξn( 1113857i 0

minus 1 if ξn( 1113857ilt 0

⎧⎪⎪⎨

⎪⎪⎩(101)

standing for the subdifferential of middot 1 As a half-space theassociated projection onto Cn takes the following form

10 Journal of Mathematics

PCn(x)

x +langξn wn minus xrang minus wn

1 + t0

ξn

2 ξn if x notin Cn

x if x isin Cn

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(102)

To show the efficiency of our algorithm we compareit with the algorithm proposed in [40] )e only dif-ference of these two algorithms is that there are no in-ertial terms in the algorithm proposed in [40] For theconvenience we denote Algorithm 1 by Algo I and thealgorithm in [40] by Algo II respectively In Algo-rithm 1 we set

θn

min 081

n2

xn minus xnminus 1

2

⎧⎨

⎩

⎫⎬

⎭ if xn ne xnminus 1

08 if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

τn

1n Alowast

Awn minus y( 1113857

if A

lowastAwn minus y( 1113857

ne 0

0 if Alowast

Awn minus y( 1113857

0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(103)

In Algo II we set θn equiv 0 and τn is chosen the same asabove )e stopping criterion is that xk+1 minus xklt ε )einitial points are x0 (0 0 0)T andx1 100(1 1 1)T )e numerical results of these twoalgorithms with different choices of the sparsity number K

are listed in Figures 1ndash4 It is easy to see that Algo Iconverges faster than Algo II does which indicates that ourmodified algorithm indeed accelerates the convergence ofthe original algorithm

106

104

102

100

||xR

ndash x R

+1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 1 Iterative results with K 50

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 2 Iterative results with K 40

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 3 Iterative results with K 30

104

102

100

||xR

ndash x R

ndash1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 4 Iterative results with K 20

Journal of Mathematics 11

5 Conclusions

In this paper we present two inertial relaxed CQ algorithmsfor solving split feasibility problems in Hilbert spaces byadopting variable step size )ese algorithms adopt the newconvex subset form and it is easy to calculate the projectionsonto these sets Furthermore step size selection in the al-gorithms does not depend on the operator norm )econvergence theorems are established under some mildconditions and a numerical experiment is given to illustratethe performance of the proposed algorithms

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was partially supported by the National NaturalScience Foundation of China (no 11771126)

References

[1] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquo Numerical Algo-rithms vol 8 no 2 pp 221ndash239 1994

[2] C Byrne ldquoA unified treatment of some iterative algorithms insignal processing and image reconstructionrdquo Inverse Prob-lems vol 20 no 1 pp 103ndash120 2004

[3] Y Censor T Elfving N Kopf and T Bortfeld ldquo)emultiple-sets split feasibility problem and its applications for inverseproblemsrdquo Inverse Problems vol 21 no 6 pp 2071ndash20842005

[4] C Byrne ldquoIterative oblique projection onto convex sets andthe split feasibility problemrdquo Inverse Problems vol 18 no 2pp 441ndash453 2002

[5] S He H Tian and H K Xu ldquo)e selective projection methodfor convex feasibility and split feasibility problemsrdquo Journal ofNonlinear and Convex Analysis vol 19 pp 1199ndash1215 2018

[6] S He Z Zhao and B Luo ldquoA relaxed self-adaptive CQ al-gorithm for the multiple-sets split feasibility problemrdquo Op-timization vol 64 no 9 pp 1907ndash1918 2015

[7] G L Acedo V Martin-Marquez F Wang and H XuldquoSolving the split feasibility problem without prior knowledgeof matrix normsrdquo Inverse Problems vol 28 Article ID 0850042012

[8] B Qu and N Xiu ldquoA new halfspace-relaxation projectionmethod for the split feasibility problemrdquo Linear Algebra andits Applications vol 428 no 5-6 pp 1218ndash1229 2008

[9] F Wang ldquoA splitting-relaxed projection method for solvingthe split feasibility problemrdquo Fixed Point gteory vol 14pp 211ndash218 2013

[10] F Wang ldquoOn the convergence of CQ algorithm with variablesteps for the split equality problemrdquo Numerical Algorithmsvol 74 no 3 pp 927ndash935 2017

[11] F Wang ldquoPolyakrsquos gradient method for split feasibilityproblem constrained by level setsrdquo Numerical Algorithmsvol 77 no 3 pp 925ndash938 2018

[12] H-K Xu ldquoA variable Krasnoselrsquoskii-Mann algorithm and themultiple-set split feasibility problemrdquo Inverse Problemsvol 22 no 6 pp 2021ndash2034 2006

[13] Q Yang ldquo)e relaxed CQ algorithm solving the split feasi-bility problemrdquo Inverse Problems vol 20 no 4pp 1261ndash1266 2004

[14] F Alvarez and H Attouch ldquoAn inertial proximal method formaximal monotone operators via discretization of a nonlinearoscillator with dampingrdquo Set-Valued Analysis vol 9 pp 3ndash112001

[15] Y Dang J Sun J Sun and H Xu ldquoInertial accelerated al-gorithms for solving a split feasibility problemrdquo Journal ofIndustrial amp Management Optimization vol 13 no 3pp 1383ndash1394 2017

[16] A Gibali D Mai D )i Mai and N )e Vinh ldquoA newrelaxed CQ algorithm for solving split feasibility problems inHilbert spaces and its applicationsrdquo Journal of Industrial ampManagement Optimization vol 15 no 2 pp 963ndash984 2019

[17] Q L Dong Y J Cho and T M Rassias ldquoGeneral inertialMann algorithms and their convergence analysis for non-expansive mappingsrdquo in Applications of Nonlinear Analysispp 175ndash191 Springer Berlin Germany 2018

[18] Q L Dong Y J Cho L L Zhong and T M Rassias ldquoInertialprojection and contraction algorithms for variational in-equalitiesrdquo Journal of Global Optimization vol 70 no 3pp 687ndash704 2018

[19] Q L Dong J Z Huang X H Li Y J Cho and T M RassiasldquoMiKM multi-step inertial Krasnoselrsquoskiǐ-Mann algorithmand its applicationsrdquo Journal of Global Optimization vol 73no 4 pp 801ndash824 2019

[20] Q L Dong H B Yuan Y J Cho and T M RassiasldquoModified inertial Mann algorithm and inertial CQ-algorithmfor nonexpansive mappingsrdquo Optimization Letters vol 12no 1 pp 87ndash102 2018

[21] Y Shehu and A Gibali ldquoNew inertial relaxed method forsolving split feasibilitiesrdquo Optimization Letters 2020

[22] S S Tai N Pholasa and P Cholamjiak ldquo)emodified inertialrelaxed CQ algorithm for solving the split feasibility prob-lemsrdquo Journal of Industrial and Management Optimizationvol 14 pp 1595ndash1615 2018

[23] S S Tai N Pholasa and P Cholamjiak ldquoRelaxed CQ algo-rithms involving the inertial technique for multiple-sets splitfeasibility problemsrdquo Revista de la Real Academia de CienciasExactas Fisicas y Naturales-Serie A Matematicas vol 113pp 1081ndash1099 2019

[24] S Y Cho ldquoA convergence theorem for generalized mixedequilibrium problems and multivalued asymptotically non-expansive mappingsrdquo Journal of Nonlinear and ConvexAnalysis vol 21 pp 1017ndash1026 2020

[25] H-K Xu ldquoIterative methods for the split feasibility problemin infinite-dimensional Hilbert spacesrdquo Inverse Problemsvol 26 no 10 Article ID 105018 2010

[26] F Wang and H Xu ldquoApproximating curve and strongconvergence of the CQ algorithm for the split feasibilityproblemrdquo Journal of Inequalities and Applications vol 2010Article ID 102085 2010

[27] M Tian and H F Zhang ldquo)e regularized CQ algorithmwithout a priori knowledge of operator norm for solving thesplit feasibility problemrdquo Journal of Inequalities and Appli-cations vol 2017 p 207 2017

[28] Y H Yao W J Gang and Y C Liou ldquoRegularized methodsfor the split feasibility problemrdquo Abstract and AppliedAnalysis vol 2012 Article ID 140679 2012

12 Journal of Mathematics

[29] Y Zhou Z Haiyun and P Wang ldquoIterative methods forfinding the minimum-norm solution of the standardmonotone variational inequality problems with applicationsin Hilbert spacesrdquo Journal of Inequalities and Applicationsvol 2015 2015

[30] T V Nguyen C Prasit and S Suthep ldquoA new CQ algorithmfor solving split feasibility problems in Hilbert spacesrdquoMalaysian Mathematical Sciences Society and Penerbit Uni-versiti Sains Malaysia vol 42 no 5 2020

[31] D R Sahu Y J Cho Q L Dong M R Kashyap and X H LiInertial Relaxed CQ algorithms for Solving a Split FeasibilityProblem in Hilbert Spaces Springer Science Business MediaBerlin Germany 2020

[32] R P Agarwal D O Regan and D R Sahu ldquoFixed pointtheory for Lipschitzian-type mappings with applicationsrdquo inTopological Fixed Point gteory and Its Applications SpringerNew York NY USA 2009

[33] S Y Cho ldquoAmonotone Bregan projection algorithm for fixedpoint and equilibrium problems in a relative Banach spacerdquoFilomat vol 34 pp 1487ndash1497 2020

[34] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator gteory in Hilbert Spaces Springer NewYork NY USA 2011

[35] P E Mainge ldquoInertial iterative process for fixed points ofcertain quasi-nonexpansive mappingrdquo Set Valued Analysisvol 15 pp 67ndash79 2007

[36] P E Mainge ldquoApproximation methods for common fixedpoints of nonexpansive mappings in Hilbert spacesrdquo Journalof Mathematical Analysis and Applications vol 325pp 469ndash479 2007

[37] A Moudafi and A Gibali ldquoregularization of split feasibilityproblemsrdquo Numerical Algorithmsvol 78 no 3 pp 1ndash192017

[38] S He and C Yang ldquoSolving the variational inequality problemdefined on intersection of finite level setsrdquo Abstract andApplied Analysis vol 2013 Article ID 942315 2013

[39] F Wang and H Yu ldquoAn inertial relaxed CQ algorithm withan application to the LASSO and elastic netrdquo Optimizationvol 2020 Article ID 1763989 2020

[40] P K Anh N T Vinh and V T Dung ldquoA new self-adaptiveCQ algorithm with an application to the LASSO problemrdquoJournal of Fixed Point gteory and Applications vol 20 p 1422018

Journal of Mathematics 13

Lemma 3 (see [34]) Let f H⟶ (minus infin +infin] be anα-strongly convex function gten for all x y isin H

f(y) gef(x) +langξ y minus xrang +α2

y minus x2 ξ isinzf(x) (27)

Lemma 4 (see [25]) Let tgt 0 and xlowast isin H gten the followingstatements are equivalent

(1) gte point xlowast solves the problem (SFP)(2) gte point xlowast solves the fixed-point equation

xlowast

PC xlowast

minus tAlowast

I minus PQ1113872 1113873Axlowast

1113872 1113873 (28)

(3) gte point xlowast solves the variational inequality problemwith respect to the gradient of f that is find a pointx isin C such that

langnablaf(x) y minus xrangge 0 forally isin C (29)

Lemma 5 (see [16]) Let H be a real Hilbert space and letxn1113864 1113865 be a sequence in H such that there exists a nonemptyclosed and convex subset S of H satisfying the followingconditions

(i) For all z isin S limn⟶infinxn minus z exists(ii) Any weak cluster point of xn1113864 1113865 belongs to S

gten there exists xlowast isin S such that xn1113864 1113865 converges weaklyto xlowast

Lemma 6 (see [35]) Let ϕn1113864 1113865 sub [0infin) and δn1113864 1113865 sub [0infin) betwo nonnegative real sequences satisfying the followingconditions

(1) ϕn+1 minus ϕn le θn(ϕn minus ϕnminus 1) + δn

(2) 1113936infinn1 δn ltinfin

(3) θn1113864 1113865 sub [0 θ] where θ isin [0 1)

gten ϕn1113864 1113865 is a converging sequence and1113936infinn1 [ϕn+1 minus ϕn]+ltinfin where [t]+ max t 0 for any t isin R

Lemma 7 (see [36 37]) Let an1113864 1113865infinn0 and cn1113864 1113865

infinn0 be sequences

of nonnegative real numbers such that

an+1 le 1 minus βn( 1113857an + δn + cn nge 1 (30)

where βn1113864 1113865infinn0 is a sequence in (0 1) and δn1113864 1113865

infinn0 is a real

sequence Assume 1113936infinn1 cn ltinfin gten the following results

hold

(1) If δn le βnM for some Mge 0 then an1113864 1113865infinn0 is a bounded

sequence(2) If 1113936

infinn1 βn infin and lim supn⟶infin(δnβn)le 0 then

limn⟶infinan 0

Lemma 8 (see [38]) Assume that sn1113864 1113865 is a sequence ofnonnegative real numbers such that

sn+1 le 1 minus αn( 1113857sn + αnδn nge 1

sn+1 le sn minus λn + cn nge 1(31)

where αn1113864 1113865 is a sequence in (0 1) λn1113864 1113865 is a sequence ofnonnegative real numbers and δn1113864 1113865 and cn1113864 1113865 are two se-quences in R such that

(1) 1113936infinn1 αn infin

(2) limn⟶infincn 0(3) limk⟶infinλnk

0 implies lim supk⟶infinδnkle 0 for any

subsequence nk1113864 1113865 of n

gten limn⟶infinsn 0

3 Convergence Analysis

In this section we consider the (SFP) in which C is given by

C x isin H1|c(x)le 01113864 1113865 (32)

where c H1⟶ (minus infin +infin] is an α-strongly convex func-tion the set Q is given by

Q y isin H2|q(y)le 01113864 1113865 (33)

where q H2⟶ (minus infin +infin] is a β-strongly convex func-tion We assume that the solution set S of the (SFP) isnonempty and c and q are lower semicontinuous convexfunctions furthermore we also assume that zc and zq arebounded operators (ie bounded on bounded sets)

We agree to build the following sets in our algorithmsaccording to [39] that is given the n-th iterative point wnwe construct Cn as

Cn x isin H1|c wn( 1113857 +langξn x minus wnrang +α2

x minus wn

2 le 01113882 1113883

(34)

where ξn isinzc(wn)

Qn y isin H2|q Awn( 1113857 +langζn y minus Awnrang +β2

y minus Awn

2 le 01113896 1113897

(35)

where ζn isinzq(Awn)If α 0 and β 0 then Cn and Qn are reduced to the

half-spaces Cn and Qn respectively If αgt 0 and βgt 0 thenCn and Qn are nonempty closed ball of radius

(1α)

ξn2 minus 2αc(wn)

1113969

centred at wn minus (1α)ξn and

(1β)

ζn2 minus 2βq(Awn)

1113969

centred at Awn minus (1β)ζnrespectively

In addition for each nge 0 we define the followingfunctions

fn(x) 12

I minus PQn1113872 1113873Ax

2

nablafn(x) Alowast

I minus PQn1113872 1113873Ax

(36)

where Qn is given as in (35) fn is weakly lower semi-continuous convex and differentiable and its gradient nablafn

4 Journal of Mathematics

is Lipschitz continuous Now we propose new relaxed CQ

algorithms for solving the (SFP)Next two inertial relaxed CQ algorithms will be in-

troduced )e weak convergence of Algorithm 1 and thestrong convergence of Algorithm 2 will be proved underdifferent step sizes

Algorithm 1 Choose positive sequence εn1113864 1113865 satisfying1113936infinn0 εn ltinfinLet x0 x1 isin C be arbitrary Given xn xnminus 1 update the

next iteration via

wn xn + θn xn minus xnminus 1( 1113857

xn+1 PCnwn minus τnnablafn wn( 1113857( 1113857

(37)

where 0le θn lt θn and

θn

min θεn

max xn minus xnminus 1

2 xn minus xnminus 1

1113882 1113883

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ if xn ne xnminus 1

θ if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(38)

and Cn and Qn are given as in (34) and (35)

τn

σn

nablafn wn( 1113857

if nablafn wn( 1113857

ne 0

0 if nablafn wn( 1113857

0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(39)

where 1113936infinn1 σn infin 1113936

infinn1 σ

2n ltinfin

If xn+1 wn then stop otherwise set n n + 1 and goto the next iteration

By assuming θn we know

1113944

infin

n1θn xn minus xnminus 1

2 ltinfin

1113944

infin

n1θn xn minus xnminus 1

ltinfin

(40)

which means

limn⟶infin

θn xn minus xnminus 1

2

0

limn⟶infin

θn xn minus xnminus 1

0(41)

From ξn isinzc(wn) applying Lemma 3 we get CsubeCn anda similar way is used to get QsubeQn

Now let us show that our proposed algorithm has a veryimportant property if xn+1 wn for some ngt 0 then wn is asolution of (SFP) Indeed xn+1 isin Cn so that wn isin Cn as wn

xn+1 by assumption So we get c(wn)le 0 from (34) that iswn isin C On the other hand according to the algorithm wehave wn PCn

(wn minus τnAlowast(I minus PQn)Awn) which together

with Lemma 4 implies that Awn isin Qn It also implies thatq(Awn)le 0 from (35) then Awn isin Q )e conclusion istenable

Lemma 9 Let xn1113864 1113865 and wn1113864 1113865 be the sequences generated byAlgorithm 1 gten for any z isin S it follows that

xn+1 minus z

2 le wn minus z

2

+ σ2n minus4σnfn wn( 1113857

nablafn wn( 1113857

(42)

Proof For z isin S we have z isin C Az isin Q and we havez PCz PCn

z Az PQAz PQnAz

It follows from Lemma 1 that

xn+1 minus z

2

PCnwn minus τnnablafn wn( 1113857( 1113857 minus z

2

le wn minus z( 1113857 minus τnnablafn wn( 1113857

2

minus wn minus xn+1( 1113857 minus τnnablafn wn( 1113857

2

wn minus z

2

minus wn minus xn+1

2

minus 2τnlangnablafn wn( 1113857 wn minus zrang + 2τnlangnablafn wn( 1113857 wn minus xn+1rang

(43)

where

2τnlangnablafn wn( 1113857 wn minus zrang 2τnlang I minus PQn1113872 1113873Awn minus I minus PQn

1113872 1113873Az Awn minus Azrang

ge 2τn I minus PQn1113872 1113873Awn

2

4τnfn wn( 1113857

2τnlangnablafn wn( 1113857 wn minus xn+1rang le 2τn nablafn wn( 1113857

middot wn minus xn+1

le wn minus xn+1

2

+ τ2n nablafn wn( 1113857

2

(44)

Journal of Mathematics 5

Hence we have

xn+1 minus z

2 le wn minus z

2

minus wn minus xn+1

2

minus 4τnfn wn( 1113857

+ wn minus xn+1

2

+ τ2n nablafn wn( 1113857

2

(45)

If nablafn(wn) 0 then τn 0 so that

xn+1 minus z

2 le wn minus z

2 (46)

If nablafn(wn)ne 0 we have

xn+1 minus z

2 le wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 4τnfn wn( 1113857

wn minus z

2

+ σ2n minus4σnfn wn( 1113857

nablafn wn( 1113857

(47)

)e proof is complete

Theorem 1 Assume that θn satisfies the assumption gten

there exists a subsequent xnj1113882 1113883 of xn1113864 1113865 generated by Algo-

rithm 1 which weakly converges to a solution of (SFP)

Proof We first show that for any z isin S the limit ofxn minus z1113864 1113865 exists By applying Lemma 9 we have

xn+1 minus z

2 le wn minus z

2

+ σ2n minus4σnfn wn( 1113857

nablafn wn( 1113857

(48)

From the construction of wn and Lemma 2 we have

wn minus z

2

1 + θn( 1113857 xn minus z( 1113857 minus θn xnminus 1 minus z( 1113857

2

1 + θn( 1113857 xn minus z

2

minus θn xnminus 1 minus z

2

+ θn 1 + θn( 1113857 xn minus xnminus 1

2

(49)

le 1 + θn( 1113857 xn minus z

2

minus θn xnminus 1 minus z

2

+ 2θn xn minus xnminus 1

2

(50)

Combining (48) and (50) immediately we get

xn+1 minus z

2 le 1 + θn( 1113857 xn minus z

2

minus θn xnminus 1 minus z

2

+ 2θn xn minus xnminus 1

2

+ σ2n(51)

Denote ϕn xn minus z2 from (51) we have

ϕn+1 minus ϕn le θn ϕn minus ϕnminus 1( 1113857 + 2θn xn minus xnminus 1

2

+ σ2n (52)

where

1113944

infin

n1θn xn minus xnminus 1

2 ltinfin

1113944

infin

n1σ2n ltinfin

(53)

Using Lemma 6 the limit of ϕn exists and1113936infinn1 (xn+1 minus z2 minus xn minus z2)+ltinfin which implies that

1113936infinn1(xn+1 minus z2 minus xn minus z2)ltinfin (xn+1 minus z2 minus xn

minus z2)+ max xn+1 minus z2 minus xn minus z2 01113966 1113967 )is also impliesthat the sequence xn1113864 1113865 is bounded so wn1113864 1113865 is bounded

We next show that ωw(xn) sub S Since wn1113864 1113865 is boundedfrom the Lipschitz continuity of nablafn we get thatnablafn(wn)1113864 1113865 is bounded From (48) and (50) we get

4σnfn wn( 1113857

nablafn wn( 1113857

le xn minus z

2

minus xn+1 minus z

2

+ θn xn minus z

2

minus xnminus 1 minus z

2

1113874 1113875 + 2θn xn minus xnminus 1

2

+ σ2n (54)

where 1113936infinn1(xn+1 minus z2 minus xn minus z2)ltinfin 1113936

infinn1 θnxnminus

xnminus 12 ltinfin and 1113936

infinn1 σ2n ltinfin so we have

1113944

infin

n1

4σnfn wn( 1113857

nablafn wn( 1113857

ltinfin (55)

But 1113936infinn1 σn infin so

lim infn⟶infin

fn wn( 1113857 0

ie lim infn⟶infin

I minus PQn1113872 1113873Awn

2

0(56)

On the other hand since xn1113864 1113865 is bounded the set ωw(xn)

is nonempty Let xlowast isin ωw(xn) then there exists a subse-quence xnk

1113966 1113967 of xn1113864 1113865 such that xnkxlowast Furthermore

wn minus xn

2

θ2n xn minus xnminus 1

2 le θn xn minus xnminus 1

2⟶ 0

(57)

Let wnj1113882 1113883 be a subsequence of the sequence wn1113864 1113865 such

that

lim infn⟶infin

I minus PQn1113872 1113873Awn

2

limj⟶infin

I minus PQnj1113874 1113875Awnj

2 0

(58)

Since wnj1113882 1113883 is bounded there exists a subsequence wnjm

1113882 1113883

of wnj1113882 1113883 which converges weakly to xlowast Without loss of

generality we can assume that wnjxlowast and A is a bounded

linear operator so AwnjAxlowast

From Lemma 1 we conclude that

langwn minus τnnablafn wn( 1113857 minus xn+1 z minus xn+1rang le 0 (59)

Since τn⟶ 0 and nablafn(wn)1113864 1113865 is bounded we haveτnnablafn(wn)⟶ 0 Hence we get

6 Journal of Mathematics

langxn+1 minus wn xn+1 minus zrang le langτnnablafn wn( 1113857 z minus xn+1rang⟶ 0

(60)

Since 1113936infinn1(xn+1 minus z2 minus xn minus z2)ltinfin and

1113936infinn1 θnxn minus xnminus 1

2 ltinfin from (50) we obtain

xn+1 minus wn

2

wn minus z

2

minus xn+1 minus z

2

+ 2langxn+1 minus wn xn+1 minus zrang

le xn minus z

2

minus xn+1 minus z

2

+ θn xn minus z

2

minus xnminus 1 minus z

2

1113874 1113875 + 2θn xn minus xnminus 1

2

+ 2langxn+1 minus wn xn+1 minus zrang⟶ 0

(61)

)us

xn+1 minus xn

le xn+1 minus wn

+ wn minus xn

⟶ 0 (62)

Since PQnjAwnjisin Qnj

by the definition of Qnj

q Awnj1113874 1113875 +langζnj

PQnj

Awnjminus Awnjrang +

β2

PQnj

Awnjminus Awnj

2le 0

(63)

where ζnjisinzq(Awnj

) From the boundedness assumption ofzq and limj⟶infin(I minus PQnj

)Awnj2 0 we have

q Awnj1113874 1113875le ζnj

middot I minus PQnj1113874 1113875Awnj

minusβ2

I minus PQnj1113874 1113875Awnj

2⟶ 0

(64)

From the weak lower semicontinuity of the convexfunction q it follows that

q Axlowast

( 1113857le lim infj⟶infin

q Awnj1113874 1113875le 0 (65)

which means that Axlowast isin QFurthermore xnj+1 isin Cnj

and by the definition of Cnj

c wnj1113874 1113875 +langξnj

xnj+1 minus wnjrang +

α2

xnj+1 minus wnj

2le 0 (66)

where ξnjisinzc(wnj

) From the boundedness assumption of zc

and xnj+1 minus wnj⟶ 0 we have

c wnj1113874 1113875le ξnj

middot wnjminus xnj+1

minusα2

xnj+1 minus wnj

2⟶ 0

(67)

From the weak lower semicontinuity of the convexfunction c it follows that

c xlowast

( 1113857le lim infj⟶infin

c wnj1113874 1113875le 0 (68)

which means that xlowast isin C )erefore xnjxlowast isin S )e proof

is complete

Algorithm 2 Choose positive sequence εn1113864 1113865 satisfying1113936infinn0 εn ltinfinLet x0 x1 isin C be arbitrary Given xn xnminus 1 update the

next iteration via

wn xn + θn xn minus xnminus 1( 1113857

xn+1 βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859

τn ρnfn wn( 1113857

nablafn wn( 1113857

2

(69)

where 0le θn lt θn and

θn

min θεn

max xn minus xnminus 1

2 xn minus xnminus 1

1113882 1113883

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ if xn nexnminus 1

θ if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(70)

and Cn and Qn are given as in (34) and (35) βn1113864 1113865 sub (0 1)limn⟶infinβn 0 1113936

infinn1 βn infin and infnρn(4 minus ρn)gt 0

If xn+1 wn then stop otherwise set n n + 1 and goto the next iteration

Theorem 2 Assume that infnρn(4 minus ρn)gt 0 and εn o(βn)gten the sequence xn generated by Algorithm 2 convergesstrongly to z PSu

Proof First we show that for any z isin S the sequence xn1113864 1113865

is bounded From the construction of wn we have

Journal of Mathematics 7

wn minus z

xn + θn xn minus xnminus 1( 1113857 minus z

le xn minus z

+ θn xn minus xnminus 1

(71)

wn minus τnnablafn wn( 1113857 minus z

2

wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 2τnlangnablafn wn( 1113857 wn minus zrang

le wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 4τnfn wn( 1113857

wn minus z

2

+ρ2nf

2n wn( 1113857

nablafn wn( 1113857

2 minus

4ρnf2n wn( 1113857

nablafn wn( 1113857

2

wn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

le wn minus z

2

(72)

So combining (71) and (72) we get

xn+1 minus z

βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus z

PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z + βn(u minus z)

le PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z

+ βnu minus z

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 1113857 minus 1 minus βn( 1113857z

+ βnu minus z

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus z

+ βnu minus z

le 1 minus βn( 1113857 wn minus z

+ βnu minus z

le 1 minus βn( 1113857 xn minus z

+ θn xn minus xnminus 1

1113960 1113961 + βnu minus z

le 1 minus βn( 1113857 xn minus z

+ βn σn +u minus z1113858 1113859

(73)

where σn (1 minus βn)(θnβn)xn minus xnminus 1 According to hy-pothesis θn

θn leεn

xn minus xnminus 1

rArrθn

βn

xn minus xnminus 1

leεn

βn

⟶ 0 (74)

Note that

limn⟶infin

σn limn⟶infin

1 minus βn( 1113857θn

βn

xn minus xnminus 1

0 (75)

which implies that the sequence σn1113864 1113865 is bounded Setting

M max supnisinN

σn u minus z1113896 1113897 (76)

as well as using Lemma 7 we conclude that the sequencexn minus z1113864 1113865 is bounded )is shows that the sequence xn1113864 1113865 isbounded and so is wn1113864 1113865

Since xn minus z1113864 1113865 is bounded assume that there exists aconstant M1 such that xn minus zleM1 )us

wn minus z

2 le xn minus z

+ θn xn minus xnminus 1

1113872 1113873

2

xn minus z

2

+ θ2n xn minus xnminus 1

2

+ 2θn xn minus xnminus 1

middot xn minus z

le xn minus z

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

(77)

and we get

8 Journal of Mathematics

xn+1 minus z

2

βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus z

2

PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z + βn(u minus z)

2

le PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z

2

+ 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 1113857 minus 1 minus βn( 1113857z

2

+ 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 wn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

⎡⎢⎢⎣ ⎤⎥⎥⎦ + 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 xn minus z

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

⎡⎢⎢⎣ ⎤⎥⎥⎦ + 2βnlangu minus z xn+1 minus zrang

(78)

From (78)

xn+1 minus z

2 le 1 minus βn( 1113857 xn minus z

2

+ βn

θn

βn

xn minus xnminus 1

2

+ 2M11113890

middotθn

βn

xn minus xnminus 1

+ 2langu minus z xn+1 minus zrang1113891

xn+1 minus z

2 le xn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang(79)

Let

sn xn minus z

2

δn θn

βn

xn minus xnminus 1

2

+ 2M1 middotθn

βn

xn minus xnminus 1

+ 2langu minus z xn+1 minus zrang

ηn ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

cn θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang

(80)

)en (78) can reduce to the inequalities

sn+1 le 1 minus βn( 1113857sn + βnδn nge 1

sn+1 le sn minus ηn + cn(81)

Furthermore we know that

1113944

infin

n0βn infin (82)

limn⟶infin

cn limn⟶infin

θn xn minus xnminus 1

2

+ 2M11113876

middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang1113961 0

(83)

Let nk1113864 1113865 be a subsequence of n and suppose that

limk⟶infin

ηnk 0 (84)

)en we have

limk⟶infin

ρnk4 minus ρnk

1113872 1113873f2nk

wnk1113872 1113873

nablafnkwnk

1113872 1113873

2 0 (85)

which implies by our assumption that

f2nk

wnk1113872 1113873

nablafnkwnk

1113872 1113873

2⟶ 0 as k⟶infin (86)

Since nablafnk(wnk

)1113966 1113967 is bounded it follows thatfnk

(wnk)⟶ 0 as k⟶infin so we get

limk⟶infin(I minus PQnk

)Awnk 0

We next show that ωw(xn) sub S Since xn1113864 1113865 is boundedthe set ωw(xn) is nonempty Let xlowast isin ωw(xn) then thereexists a subsequence xnk

1113966 1113967 of xn1113864 1113865 such that xnkxlowast

wn minus xn

xn + θn xn minus xnminus 1( 1113857 minus xn

θn xn minus xnminus 1

⟶ 0

(87)

and then wnjxlowast and A is a bounded linear operator so

AwnjAxlowast

Since PQnjAwnjisin Qnj

we have

q Awnj1113874 1113875 +langζnj

PQnj

Awnjminus Awnjrang +

β2

PQnj

Awnjminus Awnj

2le 0

(88)

Journal of Mathematics 9

where ζnjisinzq(Awnj

) and by the boundedness of zq we get

q Awnj1113874 1113875le ζnj

middot I minus PQnj1113874 1113875Awnj

minusβ2

I minus PQnj1113874 1113875Awnj

2⟶ 0

(89)

and using the weak lower semicontinuity of q

q Axlowast

( 1113857le lim infj⟶infin

q Awnj1113874 1113875le 0 (90)

)us Axlowast isin QOn the other hand

xn+1 minus wn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus wn

+ βnu minus z

1 minus βn( 1113857 middotρnfn wn( 1113857

nablafn wn( 1113857

+ βnu minus z⟶ 0

(91)

Since (xnj+1 minus βnju) isin Cnj

we have

c wnj1113874 1113875 +langξnj

xnj+1 minus βnju minus wnjrang +

α2

xnj+1 minus βnju minus wnj

2le 0

(92)

where ξnjisinzc(wnj

) and by the boundedness of zc we get

c wnj1113874 1113875le ξnj

middot wnjminus xnj+1 + βnj

u

minusα2

xnj+1 minus wnjminus βnj

u

2

le ξnj

middot wnjminus xnj+1

+ βnju1113876 1113877

minusα2

xnj+1 minus wnj

2

+ β2nju

2minus 2langxnj+1 minus wnj

βnjurang1113876 1113877⟶ 0

(93)

and using the weak lower semicontinuity of c

c xlowast

( 1113857le lim infj⟶infin

c wnj1113874 1113875le 0 (94)

)us xlowast isin C then xlowast isin S that is ωw(xn) sub SNext we have

xn+1 minus xn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus xn

+ βn u minus xn

le 1 minus βn( 1113857 wn minus xn

+ τn nablafn wn( 1113857

1113960 1113961 + βn xn minus u

le wn minus xn

+

ρnfn wn( 1113857

nablafn wn( 1113857

+ βn xn minus u

⟶ 0

(95)

For z PSu and xnkxlowast isin S using Lemma 1

langu minus z xlowast minus zrangle 0 so

lim supn⟶infinlangu minus z xn minus zrang lim sup

k⟶infinlangu minus z xnk

minus zrang

langu minus z xlowast

minus zrang le 0

(96)

and then

lim supn⟶infinlangu minus z xn+1 minus zrang

lim supn⟶infinlangu minus z xn+1 minus xnrang +langu minus z xn minus zrang( 1113857le 0

(97)

and thus

lim supk

δnk lim sup

k

θnk

βnk

xnkminus xnkminus 1

2

+ 2M11113890

middotθnk

βnk

xnkminus xnkminus 1

+ 2langu minus z xnk+1 minus zrang1113891le 0

(98)

From (82) (83) (98) and Lemma 8 we conclude that thesequence xn1113864 1113865 converges strongly to z PSu )e proof iscomplete

4 Numerical Experiments

In this section we present a numerical experiment to il-lustrate the performance of the proposed algorithms Ournumerical experiments are coded in MATLAB R2007running on personal computer with 350GHz Intel Core i3and 4GB RAM In what follows we apply our algorithms tosolve the problem of least absolute shrinkage and selectionoperator which requires solving a convex optimizationproblem as

minxisinRn

12Ax minus y

2

st x1 le t0

(99)

where A isin Rmtimesn y isin Rm and t0 gt 0 are given elements Inour experiment we first generate an m times n matrix A ran-domly by a standardized normal distribution and x is asparse signal with n elements only K of which is nonzerowhich is also generated randomly )e observation y isgenerated as y Ax )e parameters in this experiment areset with n 512 m 256 ε 10minus 4 and t0 K In thissituation it is readily seen that C x isin Rn c(x)le 0 withc(x) x1 minus t0 and Q y1113864 1113865 which in turn implies that

Cn x isin Rn langξn xrang le langξn wnrang minus wn

1 + t01113966 1113967 (100)

where ξn is defined by

ξn( 1113857i

1 if ξn( 1113857igt 0

[minus 1 1] if ξn( 1113857i 0

minus 1 if ξn( 1113857ilt 0

⎧⎪⎪⎨

⎪⎪⎩(101)

standing for the subdifferential of middot 1 As a half-space theassociated projection onto Cn takes the following form

10 Journal of Mathematics

PCn(x)

x +langξn wn minus xrang minus wn

1 + t0

ξn

2 ξn if x notin Cn

x if x isin Cn

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(102)

To show the efficiency of our algorithm we compareit with the algorithm proposed in [40] )e only dif-ference of these two algorithms is that there are no in-ertial terms in the algorithm proposed in [40] For theconvenience we denote Algorithm 1 by Algo I and thealgorithm in [40] by Algo II respectively In Algo-rithm 1 we set

θn

min 081

n2

xn minus xnminus 1

2

⎧⎨

⎩

⎫⎬

⎭ if xn ne xnminus 1

08 if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

τn

1n Alowast

Awn minus y( 1113857

if A

lowastAwn minus y( 1113857

ne 0

0 if Alowast

Awn minus y( 1113857

0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(103)

In Algo II we set θn equiv 0 and τn is chosen the same asabove )e stopping criterion is that xk+1 minus xklt ε )einitial points are x0 (0 0 0)T andx1 100(1 1 1)T )e numerical results of these twoalgorithms with different choices of the sparsity number K

are listed in Figures 1ndash4 It is easy to see that Algo Iconverges faster than Algo II does which indicates that ourmodified algorithm indeed accelerates the convergence ofthe original algorithm

106

104

102

100

||xR

ndash x R

+1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 1 Iterative results with K 50

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 2 Iterative results with K 40

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 3 Iterative results with K 30

104

102

100

||xR

ndash x R

ndash1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 4 Iterative results with K 20

Journal of Mathematics 11

5 Conclusions

In this paper we present two inertial relaxed CQ algorithmsfor solving split feasibility problems in Hilbert spaces byadopting variable step size )ese algorithms adopt the newconvex subset form and it is easy to calculate the projectionsonto these sets Furthermore step size selection in the al-gorithms does not depend on the operator norm )econvergence theorems are established under some mildconditions and a numerical experiment is given to illustratethe performance of the proposed algorithms

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was partially supported by the National NaturalScience Foundation of China (no 11771126)

References

[1] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquo Numerical Algo-rithms vol 8 no 2 pp 221ndash239 1994

[2] C Byrne ldquoA unified treatment of some iterative algorithms insignal processing and image reconstructionrdquo Inverse Prob-lems vol 20 no 1 pp 103ndash120 2004

[3] Y Censor T Elfving N Kopf and T Bortfeld ldquo)emultiple-sets split feasibility problem and its applications for inverseproblemsrdquo Inverse Problems vol 21 no 6 pp 2071ndash20842005

[4] C Byrne ldquoIterative oblique projection onto convex sets andthe split feasibility problemrdquo Inverse Problems vol 18 no 2pp 441ndash453 2002

[5] S He H Tian and H K Xu ldquo)e selective projection methodfor convex feasibility and split feasibility problemsrdquo Journal ofNonlinear and Convex Analysis vol 19 pp 1199ndash1215 2018

[6] S He Z Zhao and B Luo ldquoA relaxed self-adaptive CQ al-gorithm for the multiple-sets split feasibility problemrdquo Op-timization vol 64 no 9 pp 1907ndash1918 2015

[7] G L Acedo V Martin-Marquez F Wang and H XuldquoSolving the split feasibility problem without prior knowledgeof matrix normsrdquo Inverse Problems vol 28 Article ID 0850042012

[8] B Qu and N Xiu ldquoA new halfspace-relaxation projectionmethod for the split feasibility problemrdquo Linear Algebra andits Applications vol 428 no 5-6 pp 1218ndash1229 2008

[9] F Wang ldquoA splitting-relaxed projection method for solvingthe split feasibility problemrdquo Fixed Point gteory vol 14pp 211ndash218 2013

[10] F Wang ldquoOn the convergence of CQ algorithm with variablesteps for the split equality problemrdquo Numerical Algorithmsvol 74 no 3 pp 927ndash935 2017

[11] F Wang ldquoPolyakrsquos gradient method for split feasibilityproblem constrained by level setsrdquo Numerical Algorithmsvol 77 no 3 pp 925ndash938 2018

[12] H-K Xu ldquoA variable Krasnoselrsquoskii-Mann algorithm and themultiple-set split feasibility problemrdquo Inverse Problemsvol 22 no 6 pp 2021ndash2034 2006

[13] Q Yang ldquo)e relaxed CQ algorithm solving the split feasi-bility problemrdquo Inverse Problems vol 20 no 4pp 1261ndash1266 2004

[14] F Alvarez and H Attouch ldquoAn inertial proximal method formaximal monotone operators via discretization of a nonlinearoscillator with dampingrdquo Set-Valued Analysis vol 9 pp 3ndash112001

[15] Y Dang J Sun J Sun and H Xu ldquoInertial accelerated al-gorithms for solving a split feasibility problemrdquo Journal ofIndustrial amp Management Optimization vol 13 no 3pp 1383ndash1394 2017

[16] A Gibali D Mai D )i Mai and N )e Vinh ldquoA newrelaxed CQ algorithm for solving split feasibility problems inHilbert spaces and its applicationsrdquo Journal of Industrial ampManagement Optimization vol 15 no 2 pp 963ndash984 2019

[17] Q L Dong Y J Cho and T M Rassias ldquoGeneral inertialMann algorithms and their convergence analysis for non-expansive mappingsrdquo in Applications of Nonlinear Analysispp 175ndash191 Springer Berlin Germany 2018

[18] Q L Dong Y J Cho L L Zhong and T M Rassias ldquoInertialprojection and contraction algorithms for variational in-equalitiesrdquo Journal of Global Optimization vol 70 no 3pp 687ndash704 2018

[19] Q L Dong J Z Huang X H Li Y J Cho and T M RassiasldquoMiKM multi-step inertial Krasnoselrsquoskiǐ-Mann algorithmand its applicationsrdquo Journal of Global Optimization vol 73no 4 pp 801ndash824 2019

[20] Q L Dong H B Yuan Y J Cho and T M RassiasldquoModified inertial Mann algorithm and inertial CQ-algorithmfor nonexpansive mappingsrdquo Optimization Letters vol 12no 1 pp 87ndash102 2018

[21] Y Shehu and A Gibali ldquoNew inertial relaxed method forsolving split feasibilitiesrdquo Optimization Letters 2020

[22] S S Tai N Pholasa and P Cholamjiak ldquo)emodified inertialrelaxed CQ algorithm for solving the split feasibility prob-lemsrdquo Journal of Industrial and Management Optimizationvol 14 pp 1595ndash1615 2018

[23] S S Tai N Pholasa and P Cholamjiak ldquoRelaxed CQ algo-rithms involving the inertial technique for multiple-sets splitfeasibility problemsrdquo Revista de la Real Academia de CienciasExactas Fisicas y Naturales-Serie A Matematicas vol 113pp 1081ndash1099 2019

[24] S Y Cho ldquoA convergence theorem for generalized mixedequilibrium problems and multivalued asymptotically non-expansive mappingsrdquo Journal of Nonlinear and ConvexAnalysis vol 21 pp 1017ndash1026 2020

[25] H-K Xu ldquoIterative methods for the split feasibility problemin infinite-dimensional Hilbert spacesrdquo Inverse Problemsvol 26 no 10 Article ID 105018 2010

[26] F Wang and H Xu ldquoApproximating curve and strongconvergence of the CQ algorithm for the split feasibilityproblemrdquo Journal of Inequalities and Applications vol 2010Article ID 102085 2010

[27] M Tian and H F Zhang ldquo)e regularized CQ algorithmwithout a priori knowledge of operator norm for solving thesplit feasibility problemrdquo Journal of Inequalities and Appli-cations vol 2017 p 207 2017

[28] Y H Yao W J Gang and Y C Liou ldquoRegularized methodsfor the split feasibility problemrdquo Abstract and AppliedAnalysis vol 2012 Article ID 140679 2012

12 Journal of Mathematics

[29] Y Zhou Z Haiyun and P Wang ldquoIterative methods forfinding the minimum-norm solution of the standardmonotone variational inequality problems with applicationsin Hilbert spacesrdquo Journal of Inequalities and Applicationsvol 2015 2015

[30] T V Nguyen C Prasit and S Suthep ldquoA new CQ algorithmfor solving split feasibility problems in Hilbert spacesrdquoMalaysian Mathematical Sciences Society and Penerbit Uni-versiti Sains Malaysia vol 42 no 5 2020

[31] D R Sahu Y J Cho Q L Dong M R Kashyap and X H LiInertial Relaxed CQ algorithms for Solving a Split FeasibilityProblem in Hilbert Spaces Springer Science Business MediaBerlin Germany 2020

[32] R P Agarwal D O Regan and D R Sahu ldquoFixed pointtheory for Lipschitzian-type mappings with applicationsrdquo inTopological Fixed Point gteory and Its Applications SpringerNew York NY USA 2009

[33] S Y Cho ldquoAmonotone Bregan projection algorithm for fixedpoint and equilibrium problems in a relative Banach spacerdquoFilomat vol 34 pp 1487ndash1497 2020

[34] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator gteory in Hilbert Spaces Springer NewYork NY USA 2011

[35] P E Mainge ldquoInertial iterative process for fixed points ofcertain quasi-nonexpansive mappingrdquo Set Valued Analysisvol 15 pp 67ndash79 2007

[36] P E Mainge ldquoApproximation methods for common fixedpoints of nonexpansive mappings in Hilbert spacesrdquo Journalof Mathematical Analysis and Applications vol 325pp 469ndash479 2007

[37] A Moudafi and A Gibali ldquoregularization of split feasibilityproblemsrdquo Numerical Algorithmsvol 78 no 3 pp 1ndash192017

[38] S He and C Yang ldquoSolving the variational inequality problemdefined on intersection of finite level setsrdquo Abstract andApplied Analysis vol 2013 Article ID 942315 2013

[39] F Wang and H Yu ldquoAn inertial relaxed CQ algorithm withan application to the LASSO and elastic netrdquo Optimizationvol 2020 Article ID 1763989 2020

[40] P K Anh N T Vinh and V T Dung ldquoA new self-adaptiveCQ algorithm with an application to the LASSO problemrdquoJournal of Fixed Point gteory and Applications vol 20 p 1422018

Journal of Mathematics 13

is Lipschitz continuous Now we propose new relaxed CQ

algorithms for solving the (SFP)Next two inertial relaxed CQ algorithms will be in-

troduced )e weak convergence of Algorithm 1 and thestrong convergence of Algorithm 2 will be proved underdifferent step sizes

Algorithm 1 Choose positive sequence εn1113864 1113865 satisfying1113936infinn0 εn ltinfinLet x0 x1 isin C be arbitrary Given xn xnminus 1 update the

next iteration via

wn xn + θn xn minus xnminus 1( 1113857

xn+1 PCnwn minus τnnablafn wn( 1113857( 1113857

(37)

where 0le θn lt θn and

θn

min θεn

max xn minus xnminus 1

2 xn minus xnminus 1

1113882 1113883

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ if xn ne xnminus 1

θ if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(38)

and Cn and Qn are given as in (34) and (35)

τn

σn

nablafn wn( 1113857

if nablafn wn( 1113857

ne 0

0 if nablafn wn( 1113857

0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(39)

where 1113936infinn1 σn infin 1113936

infinn1 σ

2n ltinfin

If xn+1 wn then stop otherwise set n n + 1 and goto the next iteration

By assuming θn we know

1113944

infin

n1θn xn minus xnminus 1

2 ltinfin

1113944

infin

n1θn xn minus xnminus 1

ltinfin

(40)

which means

limn⟶infin

θn xn minus xnminus 1

2

0

limn⟶infin

θn xn minus xnminus 1

0(41)

From ξn isinzc(wn) applying Lemma 3 we get CsubeCn anda similar way is used to get QsubeQn

Now let us show that our proposed algorithm has a veryimportant property if xn+1 wn for some ngt 0 then wn is asolution of (SFP) Indeed xn+1 isin Cn so that wn isin Cn as wn

xn+1 by assumption So we get c(wn)le 0 from (34) that iswn isin C On the other hand according to the algorithm wehave wn PCn

(wn minus τnAlowast(I minus PQn)Awn) which together

with Lemma 4 implies that Awn isin Qn It also implies thatq(Awn)le 0 from (35) then Awn isin Q )e conclusion istenable

Lemma 9 Let xn1113864 1113865 and wn1113864 1113865 be the sequences generated byAlgorithm 1 gten for any z isin S it follows that

xn+1 minus z

2 le wn minus z

2

+ σ2n minus4σnfn wn( 1113857

nablafn wn( 1113857

(42)

Proof For z isin S we have z isin C Az isin Q and we havez PCz PCn

z Az PQAz PQnAz

It follows from Lemma 1 that

xn+1 minus z

2

PCnwn minus τnnablafn wn( 1113857( 1113857 minus z

2

le wn minus z( 1113857 minus τnnablafn wn( 1113857

2

minus wn minus xn+1( 1113857 minus τnnablafn wn( 1113857

2

wn minus z

2

minus wn minus xn+1

2

minus 2τnlangnablafn wn( 1113857 wn minus zrang + 2τnlangnablafn wn( 1113857 wn minus xn+1rang

(43)

where

2τnlangnablafn wn( 1113857 wn minus zrang 2τnlang I minus PQn1113872 1113873Awn minus I minus PQn

1113872 1113873Az Awn minus Azrang

ge 2τn I minus PQn1113872 1113873Awn

2

4τnfn wn( 1113857

2τnlangnablafn wn( 1113857 wn minus xn+1rang le 2τn nablafn wn( 1113857

middot wn minus xn+1

le wn minus xn+1

2

+ τ2n nablafn wn( 1113857

2

(44)

Journal of Mathematics 5

Hence we have

xn+1 minus z

2 le wn minus z

2

minus wn minus xn+1

2

minus 4τnfn wn( 1113857

+ wn minus xn+1

2

+ τ2n nablafn wn( 1113857

2

(45)

If nablafn(wn) 0 then τn 0 so that

xn+1 minus z

2 le wn minus z

2 (46)

If nablafn(wn)ne 0 we have

xn+1 minus z

2 le wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 4τnfn wn( 1113857

wn minus z

2

+ σ2n minus4σnfn wn( 1113857

nablafn wn( 1113857

(47)

)e proof is complete

Theorem 1 Assume that θn satisfies the assumption gten

there exists a subsequent xnj1113882 1113883 of xn1113864 1113865 generated by Algo-

rithm 1 which weakly converges to a solution of (SFP)

Proof We first show that for any z isin S the limit ofxn minus z1113864 1113865 exists By applying Lemma 9 we have

xn+1 minus z

2 le wn minus z

2

+ σ2n minus4σnfn wn( 1113857

nablafn wn( 1113857

(48)

From the construction of wn and Lemma 2 we have

wn minus z

2

1 + θn( 1113857 xn minus z( 1113857 minus θn xnminus 1 minus z( 1113857

2

1 + θn( 1113857 xn minus z

2

minus θn xnminus 1 minus z

2

+ θn 1 + θn( 1113857 xn minus xnminus 1

2

(49)

le 1 + θn( 1113857 xn minus z

2

minus θn xnminus 1 minus z

2

+ 2θn xn minus xnminus 1

2

(50)

Combining (48) and (50) immediately we get

xn+1 minus z

2 le 1 + θn( 1113857 xn minus z

2

minus θn xnminus 1 minus z

2

+ 2θn xn minus xnminus 1

2

+ σ2n(51)

Denote ϕn xn minus z2 from (51) we have

ϕn+1 minus ϕn le θn ϕn minus ϕnminus 1( 1113857 + 2θn xn minus xnminus 1

2

+ σ2n (52)

where

1113944

infin

n1θn xn minus xnminus 1

2 ltinfin

1113944

infin

n1σ2n ltinfin

(53)

Using Lemma 6 the limit of ϕn exists and1113936infinn1 (xn+1 minus z2 minus xn minus z2)+ltinfin which implies that

1113936infinn1(xn+1 minus z2 minus xn minus z2)ltinfin (xn+1 minus z2 minus xn

minus z2)+ max xn+1 minus z2 minus xn minus z2 01113966 1113967 )is also impliesthat the sequence xn1113864 1113865 is bounded so wn1113864 1113865 is bounded

We next show that ωw(xn) sub S Since wn1113864 1113865 is boundedfrom the Lipschitz continuity of nablafn we get thatnablafn(wn)1113864 1113865 is bounded From (48) and (50) we get

4σnfn wn( 1113857

nablafn wn( 1113857

le xn minus z

2

minus xn+1 minus z

2

+ θn xn minus z

2

minus xnminus 1 minus z

2

1113874 1113875 + 2θn xn minus xnminus 1

2

+ σ2n (54)

where 1113936infinn1(xn+1 minus z2 minus xn minus z2)ltinfin 1113936

infinn1 θnxnminus

xnminus 12 ltinfin and 1113936

infinn1 σ2n ltinfin so we have

1113944

infin

n1

4σnfn wn( 1113857

nablafn wn( 1113857

ltinfin (55)

But 1113936infinn1 σn infin so

lim infn⟶infin

fn wn( 1113857 0

ie lim infn⟶infin

I minus PQn1113872 1113873Awn

2

0(56)

On the other hand since xn1113864 1113865 is bounded the set ωw(xn)

is nonempty Let xlowast isin ωw(xn) then there exists a subse-quence xnk

1113966 1113967 of xn1113864 1113865 such that xnkxlowast Furthermore

wn minus xn

2

θ2n xn minus xnminus 1

2 le θn xn minus xnminus 1

2⟶ 0

(57)

Let wnj1113882 1113883 be a subsequence of the sequence wn1113864 1113865 such

that

lim infn⟶infin

I minus PQn1113872 1113873Awn

2

limj⟶infin

I minus PQnj1113874 1113875Awnj

2 0

(58)

Since wnj1113882 1113883 is bounded there exists a subsequence wnjm

1113882 1113883

of wnj1113882 1113883 which converges weakly to xlowast Without loss of

generality we can assume that wnjxlowast and A is a bounded

linear operator so AwnjAxlowast

From Lemma 1 we conclude that

langwn minus τnnablafn wn( 1113857 minus xn+1 z minus xn+1rang le 0 (59)

Since τn⟶ 0 and nablafn(wn)1113864 1113865 is bounded we haveτnnablafn(wn)⟶ 0 Hence we get

6 Journal of Mathematics

langxn+1 minus wn xn+1 minus zrang le langτnnablafn wn( 1113857 z minus xn+1rang⟶ 0

(60)

Since 1113936infinn1(xn+1 minus z2 minus xn minus z2)ltinfin and

1113936infinn1 θnxn minus xnminus 1

2 ltinfin from (50) we obtain

xn+1 minus wn

2

wn minus z

2

minus xn+1 minus z

2

+ 2langxn+1 minus wn xn+1 minus zrang

le xn minus z

2

minus xn+1 minus z

2

+ θn xn minus z

2

minus xnminus 1 minus z

2

1113874 1113875 + 2θn xn minus xnminus 1

2

+ 2langxn+1 minus wn xn+1 minus zrang⟶ 0

(61)

)us

xn+1 minus xn

le xn+1 minus wn

+ wn minus xn

⟶ 0 (62)

Since PQnjAwnjisin Qnj

by the definition of Qnj

q Awnj1113874 1113875 +langζnj

PQnj

Awnjminus Awnjrang +

β2

PQnj

Awnjminus Awnj

2le 0

(63)

where ζnjisinzq(Awnj

) From the boundedness assumption ofzq and limj⟶infin(I minus PQnj

)Awnj2 0 we have

q Awnj1113874 1113875le ζnj

middot I minus PQnj1113874 1113875Awnj

minusβ2

I minus PQnj1113874 1113875Awnj

2⟶ 0

(64)

From the weak lower semicontinuity of the convexfunction q it follows that

q Axlowast

( 1113857le lim infj⟶infin

q Awnj1113874 1113875le 0 (65)

which means that Axlowast isin QFurthermore xnj+1 isin Cnj

and by the definition of Cnj

c wnj1113874 1113875 +langξnj

xnj+1 minus wnjrang +

α2

xnj+1 minus wnj

2le 0 (66)

where ξnjisinzc(wnj

) From the boundedness assumption of zc

and xnj+1 minus wnj⟶ 0 we have

c wnj1113874 1113875le ξnj

middot wnjminus xnj+1

minusα2

xnj+1 minus wnj

2⟶ 0

(67)

From the weak lower semicontinuity of the convexfunction c it follows that

c xlowast

( 1113857le lim infj⟶infin

c wnj1113874 1113875le 0 (68)

which means that xlowast isin C )erefore xnjxlowast isin S )e proof

is complete

Algorithm 2 Choose positive sequence εn1113864 1113865 satisfying1113936infinn0 εn ltinfinLet x0 x1 isin C be arbitrary Given xn xnminus 1 update the

next iteration via

wn xn + θn xn minus xnminus 1( 1113857

xn+1 βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859

τn ρnfn wn( 1113857

nablafn wn( 1113857

2

(69)

where 0le θn lt θn and

θn

min θεn

max xn minus xnminus 1

2 xn minus xnminus 1

1113882 1113883

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ if xn nexnminus 1

θ if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(70)

and Cn and Qn are given as in (34) and (35) βn1113864 1113865 sub (0 1)limn⟶infinβn 0 1113936

infinn1 βn infin and infnρn(4 minus ρn)gt 0

If xn+1 wn then stop otherwise set n n + 1 and goto the next iteration

Theorem 2 Assume that infnρn(4 minus ρn)gt 0 and εn o(βn)gten the sequence xn generated by Algorithm 2 convergesstrongly to z PSu

Proof First we show that for any z isin S the sequence xn1113864 1113865

is bounded From the construction of wn we have

Journal of Mathematics 7

wn minus z

xn + θn xn minus xnminus 1( 1113857 minus z

le xn minus z

+ θn xn minus xnminus 1

(71)

wn minus τnnablafn wn( 1113857 minus z

2

wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 2τnlangnablafn wn( 1113857 wn minus zrang

le wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 4τnfn wn( 1113857

wn minus z

2

+ρ2nf

2n wn( 1113857

nablafn wn( 1113857

2 minus

4ρnf2n wn( 1113857

nablafn wn( 1113857

2

wn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

le wn minus z

2

(72)

So combining (71) and (72) we get

xn+1 minus z

βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus z

PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z + βn(u minus z)

le PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z

+ βnu minus z

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 1113857 minus 1 minus βn( 1113857z

+ βnu minus z

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus z

+ βnu minus z

le 1 minus βn( 1113857 wn minus z

+ βnu minus z

le 1 minus βn( 1113857 xn minus z

+ θn xn minus xnminus 1

1113960 1113961 + βnu minus z

le 1 minus βn( 1113857 xn minus z

+ βn σn +u minus z1113858 1113859

(73)

where σn (1 minus βn)(θnβn)xn minus xnminus 1 According to hy-pothesis θn

θn leεn

xn minus xnminus 1

rArrθn

βn

xn minus xnminus 1

leεn

βn

⟶ 0 (74)

Note that

limn⟶infin

σn limn⟶infin

1 minus βn( 1113857θn

βn

xn minus xnminus 1

0 (75)

which implies that the sequence σn1113864 1113865 is bounded Setting

M max supnisinN

σn u minus z1113896 1113897 (76)

as well as using Lemma 7 we conclude that the sequencexn minus z1113864 1113865 is bounded )is shows that the sequence xn1113864 1113865 isbounded and so is wn1113864 1113865

Since xn minus z1113864 1113865 is bounded assume that there exists aconstant M1 such that xn minus zleM1 )us

wn minus z

2 le xn minus z

+ θn xn minus xnminus 1

1113872 1113873

2

xn minus z

2

+ θ2n xn minus xnminus 1

2

+ 2θn xn minus xnminus 1

middot xn minus z

le xn minus z

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

(77)

and we get

8 Journal of Mathematics

xn+1 minus z

2

βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus z

2

PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z + βn(u minus z)

2

le PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z

2

+ 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 1113857 minus 1 minus βn( 1113857z

2

+ 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 wn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

⎡⎢⎢⎣ ⎤⎥⎥⎦ + 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 xn minus z

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

⎡⎢⎢⎣ ⎤⎥⎥⎦ + 2βnlangu minus z xn+1 minus zrang

(78)

From (78)

xn+1 minus z

2 le 1 minus βn( 1113857 xn minus z

2

+ βn

θn

βn

xn minus xnminus 1

2

+ 2M11113890

middotθn

βn

xn minus xnminus 1

+ 2langu minus z xn+1 minus zrang1113891

xn+1 minus z

2 le xn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang(79)

Let

sn xn minus z

2

δn θn

βn

xn minus xnminus 1

2

+ 2M1 middotθn

βn

xn minus xnminus 1

+ 2langu minus z xn+1 minus zrang

ηn ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

cn θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang

(80)

)en (78) can reduce to the inequalities

sn+1 le 1 minus βn( 1113857sn + βnδn nge 1

sn+1 le sn minus ηn + cn(81)

Furthermore we know that

1113944

infin

n0βn infin (82)

limn⟶infin

cn limn⟶infin

θn xn minus xnminus 1

2

+ 2M11113876

middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang1113961 0

(83)

Let nk1113864 1113865 be a subsequence of n and suppose that

limk⟶infin

ηnk 0 (84)

)en we have

limk⟶infin

ρnk4 minus ρnk

1113872 1113873f2nk

wnk1113872 1113873

nablafnkwnk

1113872 1113873

2 0 (85)

which implies by our assumption that

f2nk

wnk1113872 1113873

nablafnkwnk

1113872 1113873

2⟶ 0 as k⟶infin (86)

Since nablafnk(wnk

)1113966 1113967 is bounded it follows thatfnk

(wnk)⟶ 0 as k⟶infin so we get

limk⟶infin(I minus PQnk

)Awnk 0

We next show that ωw(xn) sub S Since xn1113864 1113865 is boundedthe set ωw(xn) is nonempty Let xlowast isin ωw(xn) then thereexists a subsequence xnk

1113966 1113967 of xn1113864 1113865 such that xnkxlowast

wn minus xn

xn + θn xn minus xnminus 1( 1113857 minus xn

θn xn minus xnminus 1

⟶ 0

(87)

and then wnjxlowast and A is a bounded linear operator so

AwnjAxlowast

Since PQnjAwnjisin Qnj

we have

q Awnj1113874 1113875 +langζnj

PQnj

Awnjminus Awnjrang +

β2

PQnj

Awnjminus Awnj

2le 0

(88)

Journal of Mathematics 9

where ζnjisinzq(Awnj

) and by the boundedness of zq we get

q Awnj1113874 1113875le ζnj

middot I minus PQnj1113874 1113875Awnj

minusβ2

I minus PQnj1113874 1113875Awnj

2⟶ 0

(89)

and using the weak lower semicontinuity of q

q Axlowast

( 1113857le lim infj⟶infin

q Awnj1113874 1113875le 0 (90)

)us Axlowast isin QOn the other hand

xn+1 minus wn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus wn

+ βnu minus z

1 minus βn( 1113857 middotρnfn wn( 1113857

nablafn wn( 1113857

+ βnu minus z⟶ 0

(91)

Since (xnj+1 minus βnju) isin Cnj

we have

c wnj1113874 1113875 +langξnj

xnj+1 minus βnju minus wnjrang +

α2

xnj+1 minus βnju minus wnj

2le 0

(92)

where ξnjisinzc(wnj

) and by the boundedness of zc we get

c wnj1113874 1113875le ξnj

middot wnjminus xnj+1 + βnj

u

minusα2

xnj+1 minus wnjminus βnj

u

2

le ξnj

middot wnjminus xnj+1

+ βnju1113876 1113877

minusα2

xnj+1 minus wnj

2

+ β2nju

2minus 2langxnj+1 minus wnj

βnjurang1113876 1113877⟶ 0

(93)

and using the weak lower semicontinuity of c

c xlowast

( 1113857le lim infj⟶infin

c wnj1113874 1113875le 0 (94)

)us xlowast isin C then xlowast isin S that is ωw(xn) sub SNext we have

xn+1 minus xn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus xn

+ βn u minus xn

le 1 minus βn( 1113857 wn minus xn

+ τn nablafn wn( 1113857

1113960 1113961 + βn xn minus u

le wn minus xn

+

ρnfn wn( 1113857

nablafn wn( 1113857

+ βn xn minus u

⟶ 0

(95)

For z PSu and xnkxlowast isin S using Lemma 1

langu minus z xlowast minus zrangle 0 so

lim supn⟶infinlangu minus z xn minus zrang lim sup

k⟶infinlangu minus z xnk

minus zrang

langu minus z xlowast

minus zrang le 0

(96)

and then

lim supn⟶infinlangu minus z xn+1 minus zrang

lim supn⟶infinlangu minus z xn+1 minus xnrang +langu minus z xn minus zrang( 1113857le 0

(97)

and thus

lim supk

δnk lim sup

k

θnk

βnk

xnkminus xnkminus 1

2

+ 2M11113890

middotθnk

βnk

xnkminus xnkminus 1

+ 2langu minus z xnk+1 minus zrang1113891le 0

(98)

From (82) (83) (98) and Lemma 8 we conclude that thesequence xn1113864 1113865 converges strongly to z PSu )e proof iscomplete

4 Numerical Experiments

In this section we present a numerical experiment to il-lustrate the performance of the proposed algorithms Ournumerical experiments are coded in MATLAB R2007running on personal computer with 350GHz Intel Core i3and 4GB RAM In what follows we apply our algorithms tosolve the problem of least absolute shrinkage and selectionoperator which requires solving a convex optimizationproblem as

minxisinRn

12Ax minus y

2

st x1 le t0

(99)

where A isin Rmtimesn y isin Rm and t0 gt 0 are given elements Inour experiment we first generate an m times n matrix A ran-domly by a standardized normal distribution and x is asparse signal with n elements only K of which is nonzerowhich is also generated randomly )e observation y isgenerated as y Ax )e parameters in this experiment areset with n 512 m 256 ε 10minus 4 and t0 K In thissituation it is readily seen that C x isin Rn c(x)le 0 withc(x) x1 minus t0 and Q y1113864 1113865 which in turn implies that

Cn x isin Rn langξn xrang le langξn wnrang minus wn

1 + t01113966 1113967 (100)

where ξn is defined by

ξn( 1113857i

1 if ξn( 1113857igt 0

[minus 1 1] if ξn( 1113857i 0

minus 1 if ξn( 1113857ilt 0

⎧⎪⎪⎨

⎪⎪⎩(101)

standing for the subdifferential of middot 1 As a half-space theassociated projection onto Cn takes the following form

10 Journal of Mathematics

PCn(x)

x +langξn wn minus xrang minus wn

1 + t0

ξn

2 ξn if x notin Cn

x if x isin Cn

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(102)

To show the efficiency of our algorithm we compareit with the algorithm proposed in [40] )e only dif-ference of these two algorithms is that there are no in-ertial terms in the algorithm proposed in [40] For theconvenience we denote Algorithm 1 by Algo I and thealgorithm in [40] by Algo II respectively In Algo-rithm 1 we set

θn

min 081

n2

xn minus xnminus 1

2

⎧⎨

⎩

⎫⎬

⎭ if xn ne xnminus 1

08 if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

τn

1n Alowast

Awn minus y( 1113857

if A

lowastAwn minus y( 1113857

ne 0

0 if Alowast

Awn minus y( 1113857

0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(103)

In Algo II we set θn equiv 0 and τn is chosen the same asabove )e stopping criterion is that xk+1 minus xklt ε )einitial points are x0 (0 0 0)T andx1 100(1 1 1)T )e numerical results of these twoalgorithms with different choices of the sparsity number K

are listed in Figures 1ndash4 It is easy to see that Algo Iconverges faster than Algo II does which indicates that ourmodified algorithm indeed accelerates the convergence ofthe original algorithm

106

104

102

100

||xR

ndash x R

+1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 1 Iterative results with K 50

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 2 Iterative results with K 40

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 3 Iterative results with K 30

104

102

100

||xR

ndash x R

ndash1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 4 Iterative results with K 20

Journal of Mathematics 11

5 Conclusions

In this paper we present two inertial relaxed CQ algorithmsfor solving split feasibility problems in Hilbert spaces byadopting variable step size )ese algorithms adopt the newconvex subset form and it is easy to calculate the projectionsonto these sets Furthermore step size selection in the al-gorithms does not depend on the operator norm )econvergence theorems are established under some mildconditions and a numerical experiment is given to illustratethe performance of the proposed algorithms

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was partially supported by the National NaturalScience Foundation of China (no 11771126)

References

[1] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquo Numerical Algo-rithms vol 8 no 2 pp 221ndash239 1994

[2] C Byrne ldquoA unified treatment of some iterative algorithms insignal processing and image reconstructionrdquo Inverse Prob-lems vol 20 no 1 pp 103ndash120 2004

[3] Y Censor T Elfving N Kopf and T Bortfeld ldquo)emultiple-sets split feasibility problem and its applications for inverseproblemsrdquo Inverse Problems vol 21 no 6 pp 2071ndash20842005

[4] C Byrne ldquoIterative oblique projection onto convex sets andthe split feasibility problemrdquo Inverse Problems vol 18 no 2pp 441ndash453 2002

[5] S He H Tian and H K Xu ldquo)e selective projection methodfor convex feasibility and split feasibility problemsrdquo Journal ofNonlinear and Convex Analysis vol 19 pp 1199ndash1215 2018

[6] S He Z Zhao and B Luo ldquoA relaxed self-adaptive CQ al-gorithm for the multiple-sets split feasibility problemrdquo Op-timization vol 64 no 9 pp 1907ndash1918 2015

[7] G L Acedo V Martin-Marquez F Wang and H XuldquoSolving the split feasibility problem without prior knowledgeof matrix normsrdquo Inverse Problems vol 28 Article ID 0850042012

[8] B Qu and N Xiu ldquoA new halfspace-relaxation projectionmethod for the split feasibility problemrdquo Linear Algebra andits Applications vol 428 no 5-6 pp 1218ndash1229 2008

[9] F Wang ldquoA splitting-relaxed projection method for solvingthe split feasibility problemrdquo Fixed Point gteory vol 14pp 211ndash218 2013

[10] F Wang ldquoOn the convergence of CQ algorithm with variablesteps for the split equality problemrdquo Numerical Algorithmsvol 74 no 3 pp 927ndash935 2017

[11] F Wang ldquoPolyakrsquos gradient method for split feasibilityproblem constrained by level setsrdquo Numerical Algorithmsvol 77 no 3 pp 925ndash938 2018

[12] H-K Xu ldquoA variable Krasnoselrsquoskii-Mann algorithm and themultiple-set split feasibility problemrdquo Inverse Problemsvol 22 no 6 pp 2021ndash2034 2006

[13] Q Yang ldquo)e relaxed CQ algorithm solving the split feasi-bility problemrdquo Inverse Problems vol 20 no 4pp 1261ndash1266 2004

[14] F Alvarez and H Attouch ldquoAn inertial proximal method formaximal monotone operators via discretization of a nonlinearoscillator with dampingrdquo Set-Valued Analysis vol 9 pp 3ndash112001

[15] Y Dang J Sun J Sun and H Xu ldquoInertial accelerated al-gorithms for solving a split feasibility problemrdquo Journal ofIndustrial amp Management Optimization vol 13 no 3pp 1383ndash1394 2017

[16] A Gibali D Mai D )i Mai and N )e Vinh ldquoA newrelaxed CQ algorithm for solving split feasibility problems inHilbert spaces and its applicationsrdquo Journal of Industrial ampManagement Optimization vol 15 no 2 pp 963ndash984 2019

[17] Q L Dong Y J Cho and T M Rassias ldquoGeneral inertialMann algorithms and their convergence analysis for non-expansive mappingsrdquo in Applications of Nonlinear Analysispp 175ndash191 Springer Berlin Germany 2018

[18] Q L Dong Y J Cho L L Zhong and T M Rassias ldquoInertialprojection and contraction algorithms for variational in-equalitiesrdquo Journal of Global Optimization vol 70 no 3pp 687ndash704 2018

[19] Q L Dong J Z Huang X H Li Y J Cho and T M RassiasldquoMiKM multi-step inertial Krasnoselrsquoskiǐ-Mann algorithmand its applicationsrdquo Journal of Global Optimization vol 73no 4 pp 801ndash824 2019

[20] Q L Dong H B Yuan Y J Cho and T M RassiasldquoModified inertial Mann algorithm and inertial CQ-algorithmfor nonexpansive mappingsrdquo Optimization Letters vol 12no 1 pp 87ndash102 2018

[21] Y Shehu and A Gibali ldquoNew inertial relaxed method forsolving split feasibilitiesrdquo Optimization Letters 2020

[22] S S Tai N Pholasa and P Cholamjiak ldquo)emodified inertialrelaxed CQ algorithm for solving the split feasibility prob-lemsrdquo Journal of Industrial and Management Optimizationvol 14 pp 1595ndash1615 2018

[23] S S Tai N Pholasa and P Cholamjiak ldquoRelaxed CQ algo-rithms involving the inertial technique for multiple-sets splitfeasibility problemsrdquo Revista de la Real Academia de CienciasExactas Fisicas y Naturales-Serie A Matematicas vol 113pp 1081ndash1099 2019

[24] S Y Cho ldquoA convergence theorem for generalized mixedequilibrium problems and multivalued asymptotically non-expansive mappingsrdquo Journal of Nonlinear and ConvexAnalysis vol 21 pp 1017ndash1026 2020

[25] H-K Xu ldquoIterative methods for the split feasibility problemin infinite-dimensional Hilbert spacesrdquo Inverse Problemsvol 26 no 10 Article ID 105018 2010

[26] F Wang and H Xu ldquoApproximating curve and strongconvergence of the CQ algorithm for the split feasibilityproblemrdquo Journal of Inequalities and Applications vol 2010Article ID 102085 2010

[27] M Tian and H F Zhang ldquo)e regularized CQ algorithmwithout a priori knowledge of operator norm for solving thesplit feasibility problemrdquo Journal of Inequalities and Appli-cations vol 2017 p 207 2017

[28] Y H Yao W J Gang and Y C Liou ldquoRegularized methodsfor the split feasibility problemrdquo Abstract and AppliedAnalysis vol 2012 Article ID 140679 2012

12 Journal of Mathematics

[29] Y Zhou Z Haiyun and P Wang ldquoIterative methods forfinding the minimum-norm solution of the standardmonotone variational inequality problems with applicationsin Hilbert spacesrdquo Journal of Inequalities and Applicationsvol 2015 2015

[30] T V Nguyen C Prasit and S Suthep ldquoA new CQ algorithmfor solving split feasibility problems in Hilbert spacesrdquoMalaysian Mathematical Sciences Society and Penerbit Uni-versiti Sains Malaysia vol 42 no 5 2020

[31] D R Sahu Y J Cho Q L Dong M R Kashyap and X H LiInertial Relaxed CQ algorithms for Solving a Split FeasibilityProblem in Hilbert Spaces Springer Science Business MediaBerlin Germany 2020

[32] R P Agarwal D O Regan and D R Sahu ldquoFixed pointtheory for Lipschitzian-type mappings with applicationsrdquo inTopological Fixed Point gteory and Its Applications SpringerNew York NY USA 2009

[33] S Y Cho ldquoAmonotone Bregan projection algorithm for fixedpoint and equilibrium problems in a relative Banach spacerdquoFilomat vol 34 pp 1487ndash1497 2020

[34] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator gteory in Hilbert Spaces Springer NewYork NY USA 2011

[35] P E Mainge ldquoInertial iterative process for fixed points ofcertain quasi-nonexpansive mappingrdquo Set Valued Analysisvol 15 pp 67ndash79 2007

[36] P E Mainge ldquoApproximation methods for common fixedpoints of nonexpansive mappings in Hilbert spacesrdquo Journalof Mathematical Analysis and Applications vol 325pp 469ndash479 2007

[37] A Moudafi and A Gibali ldquoregularization of split feasibilityproblemsrdquo Numerical Algorithmsvol 78 no 3 pp 1ndash192017

[38] S He and C Yang ldquoSolving the variational inequality problemdefined on intersection of finite level setsrdquo Abstract andApplied Analysis vol 2013 Article ID 942315 2013

[39] F Wang and H Yu ldquoAn inertial relaxed CQ algorithm withan application to the LASSO and elastic netrdquo Optimizationvol 2020 Article ID 1763989 2020

[40] P K Anh N T Vinh and V T Dung ldquoA new self-adaptiveCQ algorithm with an application to the LASSO problemrdquoJournal of Fixed Point gteory and Applications vol 20 p 1422018

Journal of Mathematics 13

Hence we have

xn+1 minus z

2 le wn minus z

2

minus wn minus xn+1

2

minus 4τnfn wn( 1113857

+ wn minus xn+1

2

+ τ2n nablafn wn( 1113857

2

(45)

If nablafn(wn) 0 then τn 0 so that

xn+1 minus z

2 le wn minus z

2 (46)

If nablafn(wn)ne 0 we have

xn+1 minus z

2 le wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 4τnfn wn( 1113857

wn minus z

2

+ σ2n minus4σnfn wn( 1113857

nablafn wn( 1113857

(47)

)e proof is complete

Theorem 1 Assume that θn satisfies the assumption gten

there exists a subsequent xnj1113882 1113883 of xn1113864 1113865 generated by Algo-

rithm 1 which weakly converges to a solution of (SFP)

Proof We first show that for any z isin S the limit ofxn minus z1113864 1113865 exists By applying Lemma 9 we have

xn+1 minus z

2 le wn minus z

2

+ σ2n minus4σnfn wn( 1113857

nablafn wn( 1113857

(48)

From the construction of wn and Lemma 2 we have

wn minus z

2

1 + θn( 1113857 xn minus z( 1113857 minus θn xnminus 1 minus z( 1113857

2

1 + θn( 1113857 xn minus z

2

minus θn xnminus 1 minus z

2

+ θn 1 + θn( 1113857 xn minus xnminus 1

2

(49)

le 1 + θn( 1113857 xn minus z

2

minus θn xnminus 1 minus z

2

+ 2θn xn minus xnminus 1

2

(50)

Combining (48) and (50) immediately we get

xn+1 minus z

2 le 1 + θn( 1113857 xn minus z

2

minus θn xnminus 1 minus z

2

+ 2θn xn minus xnminus 1

2

+ σ2n(51)

Denote ϕn xn minus z2 from (51) we have

ϕn+1 minus ϕn le θn ϕn minus ϕnminus 1( 1113857 + 2θn xn minus xnminus 1

2

+ σ2n (52)

where

1113944

infin

n1θn xn minus xnminus 1

2 ltinfin

1113944

infin

n1σ2n ltinfin

(53)

Using Lemma 6 the limit of ϕn exists and1113936infinn1 (xn+1 minus z2 minus xn minus z2)+ltinfin which implies that

1113936infinn1(xn+1 minus z2 minus xn minus z2)ltinfin (xn+1 minus z2 minus xn

minus z2)+ max xn+1 minus z2 minus xn minus z2 01113966 1113967 )is also impliesthat the sequence xn1113864 1113865 is bounded so wn1113864 1113865 is bounded

We next show that ωw(xn) sub S Since wn1113864 1113865 is boundedfrom the Lipschitz continuity of nablafn we get thatnablafn(wn)1113864 1113865 is bounded From (48) and (50) we get

4σnfn wn( 1113857

nablafn wn( 1113857

le xn minus z

2

minus xn+1 minus z

2

+ θn xn minus z

2

minus xnminus 1 minus z

2

1113874 1113875 + 2θn xn minus xnminus 1

2

+ σ2n (54)

where 1113936infinn1(xn+1 minus z2 minus xn minus z2)ltinfin 1113936

infinn1 θnxnminus

xnminus 12 ltinfin and 1113936

infinn1 σ2n ltinfin so we have

1113944

infin

n1

4σnfn wn( 1113857

nablafn wn( 1113857

ltinfin (55)

But 1113936infinn1 σn infin so

lim infn⟶infin

fn wn( 1113857 0

ie lim infn⟶infin

I minus PQn1113872 1113873Awn

2

0(56)

On the other hand since xn1113864 1113865 is bounded the set ωw(xn)

is nonempty Let xlowast isin ωw(xn) then there exists a subse-quence xnk

1113966 1113967 of xn1113864 1113865 such that xnkxlowast Furthermore

wn minus xn

2

θ2n xn minus xnminus 1

2 le θn xn minus xnminus 1

2⟶ 0

(57)

Let wnj1113882 1113883 be a subsequence of the sequence wn1113864 1113865 such

that

lim infn⟶infin

I minus PQn1113872 1113873Awn

2

limj⟶infin

I minus PQnj1113874 1113875Awnj

2 0

(58)

Since wnj1113882 1113883 is bounded there exists a subsequence wnjm

1113882 1113883

of wnj1113882 1113883 which converges weakly to xlowast Without loss of

generality we can assume that wnjxlowast and A is a bounded

linear operator so AwnjAxlowast

From Lemma 1 we conclude that

langwn minus τnnablafn wn( 1113857 minus xn+1 z minus xn+1rang le 0 (59)

Since τn⟶ 0 and nablafn(wn)1113864 1113865 is bounded we haveτnnablafn(wn)⟶ 0 Hence we get

6 Journal of Mathematics

langxn+1 minus wn xn+1 minus zrang le langτnnablafn wn( 1113857 z minus xn+1rang⟶ 0

(60)

Since 1113936infinn1(xn+1 minus z2 minus xn minus z2)ltinfin and

1113936infinn1 θnxn minus xnminus 1

2 ltinfin from (50) we obtain

xn+1 minus wn

2

wn minus z

2

minus xn+1 minus z

2

+ 2langxn+1 minus wn xn+1 minus zrang

le xn minus z

2

minus xn+1 minus z

2

+ θn xn minus z

2

minus xnminus 1 minus z

2

1113874 1113875 + 2θn xn minus xnminus 1

2

+ 2langxn+1 minus wn xn+1 minus zrang⟶ 0

(61)

)us

xn+1 minus xn

le xn+1 minus wn

+ wn minus xn

⟶ 0 (62)

Since PQnjAwnjisin Qnj

by the definition of Qnj

q Awnj1113874 1113875 +langζnj

PQnj

Awnjminus Awnjrang +

β2

PQnj

Awnjminus Awnj

2le 0

(63)

where ζnjisinzq(Awnj

) From the boundedness assumption ofzq and limj⟶infin(I minus PQnj

)Awnj2 0 we have

q Awnj1113874 1113875le ζnj

middot I minus PQnj1113874 1113875Awnj

minusβ2

I minus PQnj1113874 1113875Awnj

2⟶ 0

(64)

From the weak lower semicontinuity of the convexfunction q it follows that

q Axlowast

( 1113857le lim infj⟶infin

q Awnj1113874 1113875le 0 (65)

which means that Axlowast isin QFurthermore xnj+1 isin Cnj

and by the definition of Cnj

c wnj1113874 1113875 +langξnj

xnj+1 minus wnjrang +

α2

xnj+1 minus wnj

2le 0 (66)

where ξnjisinzc(wnj

) From the boundedness assumption of zc

and xnj+1 minus wnj⟶ 0 we have

c wnj1113874 1113875le ξnj

middot wnjminus xnj+1

minusα2

xnj+1 minus wnj

2⟶ 0

(67)

From the weak lower semicontinuity of the convexfunction c it follows that

c xlowast

( 1113857le lim infj⟶infin

c wnj1113874 1113875le 0 (68)

which means that xlowast isin C )erefore xnjxlowast isin S )e proof

is complete

Algorithm 2 Choose positive sequence εn1113864 1113865 satisfying1113936infinn0 εn ltinfinLet x0 x1 isin C be arbitrary Given xn xnminus 1 update the

next iteration via

wn xn + θn xn minus xnminus 1( 1113857

xn+1 βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859

τn ρnfn wn( 1113857

nablafn wn( 1113857

2

(69)

where 0le θn lt θn and

θn

min θεn

max xn minus xnminus 1

2 xn minus xnminus 1

1113882 1113883

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ if xn nexnminus 1

θ if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(70)

and Cn and Qn are given as in (34) and (35) βn1113864 1113865 sub (0 1)limn⟶infinβn 0 1113936

infinn1 βn infin and infnρn(4 minus ρn)gt 0

If xn+1 wn then stop otherwise set n n + 1 and goto the next iteration

Theorem 2 Assume that infnρn(4 minus ρn)gt 0 and εn o(βn)gten the sequence xn generated by Algorithm 2 convergesstrongly to z PSu

Proof First we show that for any z isin S the sequence xn1113864 1113865

is bounded From the construction of wn we have

Journal of Mathematics 7

wn minus z

xn + θn xn minus xnminus 1( 1113857 minus z

le xn minus z

+ θn xn minus xnminus 1

(71)

wn minus τnnablafn wn( 1113857 minus z

2

wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 2τnlangnablafn wn( 1113857 wn minus zrang

le wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 4τnfn wn( 1113857

wn minus z

2

+ρ2nf

2n wn( 1113857

nablafn wn( 1113857

2 minus

4ρnf2n wn( 1113857

nablafn wn( 1113857

2

wn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

le wn minus z

2

(72)

So combining (71) and (72) we get

xn+1 minus z

βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus z

PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z + βn(u minus z)

le PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z

+ βnu minus z

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 1113857 minus 1 minus βn( 1113857z

+ βnu minus z

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus z

+ βnu minus z

le 1 minus βn( 1113857 wn minus z

+ βnu minus z

le 1 minus βn( 1113857 xn minus z

+ θn xn minus xnminus 1

1113960 1113961 + βnu minus z

le 1 minus βn( 1113857 xn minus z

+ βn σn +u minus z1113858 1113859

(73)

where σn (1 minus βn)(θnβn)xn minus xnminus 1 According to hy-pothesis θn

θn leεn

xn minus xnminus 1

rArrθn

βn

xn minus xnminus 1

leεn

βn

⟶ 0 (74)

Note that

limn⟶infin

σn limn⟶infin

1 minus βn( 1113857θn

βn

xn minus xnminus 1

0 (75)

which implies that the sequence σn1113864 1113865 is bounded Setting

M max supnisinN

σn u minus z1113896 1113897 (76)

as well as using Lemma 7 we conclude that the sequencexn minus z1113864 1113865 is bounded )is shows that the sequence xn1113864 1113865 isbounded and so is wn1113864 1113865

Since xn minus z1113864 1113865 is bounded assume that there exists aconstant M1 such that xn minus zleM1 )us

wn minus z

2 le xn minus z

+ θn xn minus xnminus 1

1113872 1113873

2

xn minus z

2

+ θ2n xn minus xnminus 1

2

+ 2θn xn minus xnminus 1

middot xn minus z

le xn minus z

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

(77)

and we get

8 Journal of Mathematics

xn+1 minus z

2

βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus z

2

PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z + βn(u minus z)

2

le PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z

2

+ 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 1113857 minus 1 minus βn( 1113857z

2

+ 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 wn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

⎡⎢⎢⎣ ⎤⎥⎥⎦ + 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 xn minus z

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

⎡⎢⎢⎣ ⎤⎥⎥⎦ + 2βnlangu minus z xn+1 minus zrang

(78)

From (78)

xn+1 minus z

2 le 1 minus βn( 1113857 xn minus z

2

+ βn

θn

βn

xn minus xnminus 1

2

+ 2M11113890

middotθn

βn

xn minus xnminus 1

+ 2langu minus z xn+1 minus zrang1113891

xn+1 minus z

2 le xn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang(79)

Let

sn xn minus z

2

δn θn

βn

xn minus xnminus 1

2

+ 2M1 middotθn

βn

xn minus xnminus 1

+ 2langu minus z xn+1 minus zrang

ηn ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

cn θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang

(80)

)en (78) can reduce to the inequalities

sn+1 le 1 minus βn( 1113857sn + βnδn nge 1

sn+1 le sn minus ηn + cn(81)

Furthermore we know that

1113944

infin

n0βn infin (82)

limn⟶infin

cn limn⟶infin

θn xn minus xnminus 1

2

+ 2M11113876

middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang1113961 0

(83)

Let nk1113864 1113865 be a subsequence of n and suppose that

limk⟶infin

ηnk 0 (84)

)en we have

limk⟶infin

ρnk4 minus ρnk

1113872 1113873f2nk

wnk1113872 1113873

nablafnkwnk

1113872 1113873

2 0 (85)

which implies by our assumption that

f2nk

wnk1113872 1113873

nablafnkwnk

1113872 1113873

2⟶ 0 as k⟶infin (86)

Since nablafnk(wnk

)1113966 1113967 is bounded it follows thatfnk

(wnk)⟶ 0 as k⟶infin so we get

limk⟶infin(I minus PQnk

)Awnk 0

We next show that ωw(xn) sub S Since xn1113864 1113865 is boundedthe set ωw(xn) is nonempty Let xlowast isin ωw(xn) then thereexists a subsequence xnk

1113966 1113967 of xn1113864 1113865 such that xnkxlowast

wn minus xn

xn + θn xn minus xnminus 1( 1113857 minus xn

θn xn minus xnminus 1

⟶ 0

(87)

and then wnjxlowast and A is a bounded linear operator so

AwnjAxlowast

Since PQnjAwnjisin Qnj

we have

q Awnj1113874 1113875 +langζnj

PQnj

Awnjminus Awnjrang +

β2

PQnj

Awnjminus Awnj

2le 0

(88)

Journal of Mathematics 9

where ζnjisinzq(Awnj

) and by the boundedness of zq we get

q Awnj1113874 1113875le ζnj

middot I minus PQnj1113874 1113875Awnj

minusβ2

I minus PQnj1113874 1113875Awnj

2⟶ 0

(89)

and using the weak lower semicontinuity of q

q Axlowast

( 1113857le lim infj⟶infin

q Awnj1113874 1113875le 0 (90)

)us Axlowast isin QOn the other hand

xn+1 minus wn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus wn

+ βnu minus z

1 minus βn( 1113857 middotρnfn wn( 1113857

nablafn wn( 1113857

+ βnu minus z⟶ 0

(91)

Since (xnj+1 minus βnju) isin Cnj

we have

c wnj1113874 1113875 +langξnj

xnj+1 minus βnju minus wnjrang +

α2

xnj+1 minus βnju minus wnj

2le 0

(92)

where ξnjisinzc(wnj

) and by the boundedness of zc we get

c wnj1113874 1113875le ξnj

middot wnjminus xnj+1 + βnj

u

minusα2

xnj+1 minus wnjminus βnj

u

2

le ξnj

middot wnjminus xnj+1

+ βnju1113876 1113877

minusα2

xnj+1 minus wnj

2

+ β2nju

2minus 2langxnj+1 minus wnj

βnjurang1113876 1113877⟶ 0

(93)

and using the weak lower semicontinuity of c

c xlowast

( 1113857le lim infj⟶infin

c wnj1113874 1113875le 0 (94)

)us xlowast isin C then xlowast isin S that is ωw(xn) sub SNext we have

xn+1 minus xn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus xn

+ βn u minus xn

le 1 minus βn( 1113857 wn minus xn

+ τn nablafn wn( 1113857

1113960 1113961 + βn xn minus u

le wn minus xn

+

ρnfn wn( 1113857

nablafn wn( 1113857

+ βn xn minus u

⟶ 0

(95)

For z PSu and xnkxlowast isin S using Lemma 1

langu minus z xlowast minus zrangle 0 so

lim supn⟶infinlangu minus z xn minus zrang lim sup

k⟶infinlangu minus z xnk

minus zrang

langu minus z xlowast

minus zrang le 0

(96)

and then

lim supn⟶infinlangu minus z xn+1 minus zrang

lim supn⟶infinlangu minus z xn+1 minus xnrang +langu minus z xn minus zrang( 1113857le 0

(97)

and thus

lim supk

δnk lim sup

k

θnk

βnk

xnkminus xnkminus 1

2

+ 2M11113890

middotθnk

βnk

xnkminus xnkminus 1

+ 2langu minus z xnk+1 minus zrang1113891le 0

(98)

From (82) (83) (98) and Lemma 8 we conclude that thesequence xn1113864 1113865 converges strongly to z PSu )e proof iscomplete

4 Numerical Experiments

In this section we present a numerical experiment to il-lustrate the performance of the proposed algorithms Ournumerical experiments are coded in MATLAB R2007running on personal computer with 350GHz Intel Core i3and 4GB RAM In what follows we apply our algorithms tosolve the problem of least absolute shrinkage and selectionoperator which requires solving a convex optimizationproblem as

minxisinRn

12Ax minus y

2

st x1 le t0

(99)

where A isin Rmtimesn y isin Rm and t0 gt 0 are given elements Inour experiment we first generate an m times n matrix A ran-domly by a standardized normal distribution and x is asparse signal with n elements only K of which is nonzerowhich is also generated randomly )e observation y isgenerated as y Ax )e parameters in this experiment areset with n 512 m 256 ε 10minus 4 and t0 K In thissituation it is readily seen that C x isin Rn c(x)le 0 withc(x) x1 minus t0 and Q y1113864 1113865 which in turn implies that

Cn x isin Rn langξn xrang le langξn wnrang minus wn

1 + t01113966 1113967 (100)

where ξn is defined by

ξn( 1113857i

1 if ξn( 1113857igt 0

[minus 1 1] if ξn( 1113857i 0

minus 1 if ξn( 1113857ilt 0

⎧⎪⎪⎨

⎪⎪⎩(101)

standing for the subdifferential of middot 1 As a half-space theassociated projection onto Cn takes the following form

10 Journal of Mathematics

PCn(x)

x +langξn wn minus xrang minus wn

1 + t0

ξn

2 ξn if x notin Cn

x if x isin Cn

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(102)

To show the efficiency of our algorithm we compareit with the algorithm proposed in [40] )e only dif-ference of these two algorithms is that there are no in-ertial terms in the algorithm proposed in [40] For theconvenience we denote Algorithm 1 by Algo I and thealgorithm in [40] by Algo II respectively In Algo-rithm 1 we set

θn

min 081

n2

xn minus xnminus 1

2

⎧⎨

⎩

⎫⎬

⎭ if xn ne xnminus 1

08 if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

τn

1n Alowast

Awn minus y( 1113857

if A

lowastAwn minus y( 1113857

ne 0

0 if Alowast

Awn minus y( 1113857

0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(103)

In Algo II we set θn equiv 0 and τn is chosen the same asabove )e stopping criterion is that xk+1 minus xklt ε )einitial points are x0 (0 0 0)T andx1 100(1 1 1)T )e numerical results of these twoalgorithms with different choices of the sparsity number K

are listed in Figures 1ndash4 It is easy to see that Algo Iconverges faster than Algo II does which indicates that ourmodified algorithm indeed accelerates the convergence ofthe original algorithm

106

104

102

100

||xR

ndash x R

+1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 1 Iterative results with K 50

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 2 Iterative results with K 40

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 3 Iterative results with K 30

104

102

100

||xR

ndash x R

ndash1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 4 Iterative results with K 20

Journal of Mathematics 11

5 Conclusions

In this paper we present two inertial relaxed CQ algorithmsfor solving split feasibility problems in Hilbert spaces byadopting variable step size )ese algorithms adopt the newconvex subset form and it is easy to calculate the projectionsonto these sets Furthermore step size selection in the al-gorithms does not depend on the operator norm )econvergence theorems are established under some mildconditions and a numerical experiment is given to illustratethe performance of the proposed algorithms

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was partially supported by the National NaturalScience Foundation of China (no 11771126)

References

[1] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquo Numerical Algo-rithms vol 8 no 2 pp 221ndash239 1994

[2] C Byrne ldquoA unified treatment of some iterative algorithms insignal processing and image reconstructionrdquo Inverse Prob-lems vol 20 no 1 pp 103ndash120 2004

[3] Y Censor T Elfving N Kopf and T Bortfeld ldquo)emultiple-sets split feasibility problem and its applications for inverseproblemsrdquo Inverse Problems vol 21 no 6 pp 2071ndash20842005

[4] C Byrne ldquoIterative oblique projection onto convex sets andthe split feasibility problemrdquo Inverse Problems vol 18 no 2pp 441ndash453 2002

[5] S He H Tian and H K Xu ldquo)e selective projection methodfor convex feasibility and split feasibility problemsrdquo Journal ofNonlinear and Convex Analysis vol 19 pp 1199ndash1215 2018

[6] S He Z Zhao and B Luo ldquoA relaxed self-adaptive CQ al-gorithm for the multiple-sets split feasibility problemrdquo Op-timization vol 64 no 9 pp 1907ndash1918 2015

[7] G L Acedo V Martin-Marquez F Wang and H XuldquoSolving the split feasibility problem without prior knowledgeof matrix normsrdquo Inverse Problems vol 28 Article ID 0850042012

[8] B Qu and N Xiu ldquoA new halfspace-relaxation projectionmethod for the split feasibility problemrdquo Linear Algebra andits Applications vol 428 no 5-6 pp 1218ndash1229 2008

[9] F Wang ldquoA splitting-relaxed projection method for solvingthe split feasibility problemrdquo Fixed Point gteory vol 14pp 211ndash218 2013

[10] F Wang ldquoOn the convergence of CQ algorithm with variablesteps for the split equality problemrdquo Numerical Algorithmsvol 74 no 3 pp 927ndash935 2017

[11] F Wang ldquoPolyakrsquos gradient method for split feasibilityproblem constrained by level setsrdquo Numerical Algorithmsvol 77 no 3 pp 925ndash938 2018

[12] H-K Xu ldquoA variable Krasnoselrsquoskii-Mann algorithm and themultiple-set split feasibility problemrdquo Inverse Problemsvol 22 no 6 pp 2021ndash2034 2006

[13] Q Yang ldquo)e relaxed CQ algorithm solving the split feasi-bility problemrdquo Inverse Problems vol 20 no 4pp 1261ndash1266 2004

[14] F Alvarez and H Attouch ldquoAn inertial proximal method formaximal monotone operators via discretization of a nonlinearoscillator with dampingrdquo Set-Valued Analysis vol 9 pp 3ndash112001

[15] Y Dang J Sun J Sun and H Xu ldquoInertial accelerated al-gorithms for solving a split feasibility problemrdquo Journal ofIndustrial amp Management Optimization vol 13 no 3pp 1383ndash1394 2017

[16] A Gibali D Mai D )i Mai and N )e Vinh ldquoA newrelaxed CQ algorithm for solving split feasibility problems inHilbert spaces and its applicationsrdquo Journal of Industrial ampManagement Optimization vol 15 no 2 pp 963ndash984 2019

[17] Q L Dong Y J Cho and T M Rassias ldquoGeneral inertialMann algorithms and their convergence analysis for non-expansive mappingsrdquo in Applications of Nonlinear Analysispp 175ndash191 Springer Berlin Germany 2018

[18] Q L Dong Y J Cho L L Zhong and T M Rassias ldquoInertialprojection and contraction algorithms for variational in-equalitiesrdquo Journal of Global Optimization vol 70 no 3pp 687ndash704 2018

[19] Q L Dong J Z Huang X H Li Y J Cho and T M RassiasldquoMiKM multi-step inertial Krasnoselrsquoskiǐ-Mann algorithmand its applicationsrdquo Journal of Global Optimization vol 73no 4 pp 801ndash824 2019

[20] Q L Dong H B Yuan Y J Cho and T M RassiasldquoModified inertial Mann algorithm and inertial CQ-algorithmfor nonexpansive mappingsrdquo Optimization Letters vol 12no 1 pp 87ndash102 2018

[21] Y Shehu and A Gibali ldquoNew inertial relaxed method forsolving split feasibilitiesrdquo Optimization Letters 2020

[22] S S Tai N Pholasa and P Cholamjiak ldquo)emodified inertialrelaxed CQ algorithm for solving the split feasibility prob-lemsrdquo Journal of Industrial and Management Optimizationvol 14 pp 1595ndash1615 2018

[23] S S Tai N Pholasa and P Cholamjiak ldquoRelaxed CQ algo-rithms involving the inertial technique for multiple-sets splitfeasibility problemsrdquo Revista de la Real Academia de CienciasExactas Fisicas y Naturales-Serie A Matematicas vol 113pp 1081ndash1099 2019

[24] S Y Cho ldquoA convergence theorem for generalized mixedequilibrium problems and multivalued asymptotically non-expansive mappingsrdquo Journal of Nonlinear and ConvexAnalysis vol 21 pp 1017ndash1026 2020

[25] H-K Xu ldquoIterative methods for the split feasibility problemin infinite-dimensional Hilbert spacesrdquo Inverse Problemsvol 26 no 10 Article ID 105018 2010

[26] F Wang and H Xu ldquoApproximating curve and strongconvergence of the CQ algorithm for the split feasibilityproblemrdquo Journal of Inequalities and Applications vol 2010Article ID 102085 2010

[27] M Tian and H F Zhang ldquo)e regularized CQ algorithmwithout a priori knowledge of operator norm for solving thesplit feasibility problemrdquo Journal of Inequalities and Appli-cations vol 2017 p 207 2017

[28] Y H Yao W J Gang and Y C Liou ldquoRegularized methodsfor the split feasibility problemrdquo Abstract and AppliedAnalysis vol 2012 Article ID 140679 2012

12 Journal of Mathematics

[29] Y Zhou Z Haiyun and P Wang ldquoIterative methods forfinding the minimum-norm solution of the standardmonotone variational inequality problems with applicationsin Hilbert spacesrdquo Journal of Inequalities and Applicationsvol 2015 2015

[30] T V Nguyen C Prasit and S Suthep ldquoA new CQ algorithmfor solving split feasibility problems in Hilbert spacesrdquoMalaysian Mathematical Sciences Society and Penerbit Uni-versiti Sains Malaysia vol 42 no 5 2020

[31] D R Sahu Y J Cho Q L Dong M R Kashyap and X H LiInertial Relaxed CQ algorithms for Solving a Split FeasibilityProblem in Hilbert Spaces Springer Science Business MediaBerlin Germany 2020

[32] R P Agarwal D O Regan and D R Sahu ldquoFixed pointtheory for Lipschitzian-type mappings with applicationsrdquo inTopological Fixed Point gteory and Its Applications SpringerNew York NY USA 2009

[33] S Y Cho ldquoAmonotone Bregan projection algorithm for fixedpoint and equilibrium problems in a relative Banach spacerdquoFilomat vol 34 pp 1487ndash1497 2020

[34] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator gteory in Hilbert Spaces Springer NewYork NY USA 2011

[35] P E Mainge ldquoInertial iterative process for fixed points ofcertain quasi-nonexpansive mappingrdquo Set Valued Analysisvol 15 pp 67ndash79 2007

[36] P E Mainge ldquoApproximation methods for common fixedpoints of nonexpansive mappings in Hilbert spacesrdquo Journalof Mathematical Analysis and Applications vol 325pp 469ndash479 2007

[37] A Moudafi and A Gibali ldquoregularization of split feasibilityproblemsrdquo Numerical Algorithmsvol 78 no 3 pp 1ndash192017

[38] S He and C Yang ldquoSolving the variational inequality problemdefined on intersection of finite level setsrdquo Abstract andApplied Analysis vol 2013 Article ID 942315 2013

[39] F Wang and H Yu ldquoAn inertial relaxed CQ algorithm withan application to the LASSO and elastic netrdquo Optimizationvol 2020 Article ID 1763989 2020

[40] P K Anh N T Vinh and V T Dung ldquoA new self-adaptiveCQ algorithm with an application to the LASSO problemrdquoJournal of Fixed Point gteory and Applications vol 20 p 1422018

Journal of Mathematics 13

langxn+1 minus wn xn+1 minus zrang le langτnnablafn wn( 1113857 z minus xn+1rang⟶ 0

(60)

Since 1113936infinn1(xn+1 minus z2 minus xn minus z2)ltinfin and

1113936infinn1 θnxn minus xnminus 1

2 ltinfin from (50) we obtain

xn+1 minus wn

2

wn minus z

2

minus xn+1 minus z

2

+ 2langxn+1 minus wn xn+1 minus zrang

le xn minus z

2

minus xn+1 minus z

2

+ θn xn minus z

2

minus xnminus 1 minus z

2

1113874 1113875 + 2θn xn minus xnminus 1

2

+ 2langxn+1 minus wn xn+1 minus zrang⟶ 0

(61)

)us

xn+1 minus xn

le xn+1 minus wn

+ wn minus xn

⟶ 0 (62)

Since PQnjAwnjisin Qnj

by the definition of Qnj

q Awnj1113874 1113875 +langζnj

PQnj

Awnjminus Awnjrang +

β2

PQnj

Awnjminus Awnj

2le 0

(63)

where ζnjisinzq(Awnj

) From the boundedness assumption ofzq and limj⟶infin(I minus PQnj

)Awnj2 0 we have

q Awnj1113874 1113875le ζnj

middot I minus PQnj1113874 1113875Awnj

minusβ2

I minus PQnj1113874 1113875Awnj

2⟶ 0

(64)

From the weak lower semicontinuity of the convexfunction q it follows that

q Axlowast

( 1113857le lim infj⟶infin

q Awnj1113874 1113875le 0 (65)

which means that Axlowast isin QFurthermore xnj+1 isin Cnj

and by the definition of Cnj

c wnj1113874 1113875 +langξnj

xnj+1 minus wnjrang +

α2

xnj+1 minus wnj

2le 0 (66)

where ξnjisinzc(wnj

) From the boundedness assumption of zc

and xnj+1 minus wnj⟶ 0 we have

c wnj1113874 1113875le ξnj

middot wnjminus xnj+1

minusα2

xnj+1 minus wnj

2⟶ 0

(67)

From the weak lower semicontinuity of the convexfunction c it follows that

c xlowast

( 1113857le lim infj⟶infin

c wnj1113874 1113875le 0 (68)

which means that xlowast isin C )erefore xnjxlowast isin S )e proof

is complete

Algorithm 2 Choose positive sequence εn1113864 1113865 satisfying1113936infinn0 εn ltinfinLet x0 x1 isin C be arbitrary Given xn xnminus 1 update the

next iteration via

wn xn + θn xn minus xnminus 1( 1113857

xn+1 βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859

τn ρnfn wn( 1113857

nablafn wn( 1113857

2

(69)

where 0le θn lt θn and

θn

min θεn

max xn minus xnminus 1

2 xn minus xnminus 1

1113882 1113883

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ if xn nexnminus 1

θ if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(70)

and Cn and Qn are given as in (34) and (35) βn1113864 1113865 sub (0 1)limn⟶infinβn 0 1113936

infinn1 βn infin and infnρn(4 minus ρn)gt 0

If xn+1 wn then stop otherwise set n n + 1 and goto the next iteration

Theorem 2 Assume that infnρn(4 minus ρn)gt 0 and εn o(βn)gten the sequence xn generated by Algorithm 2 convergesstrongly to z PSu

Proof First we show that for any z isin S the sequence xn1113864 1113865

is bounded From the construction of wn we have

Journal of Mathematics 7

wn minus z

xn + θn xn minus xnminus 1( 1113857 minus z

le xn minus z

+ θn xn minus xnminus 1

(71)

wn minus τnnablafn wn( 1113857 minus z

2

wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 2τnlangnablafn wn( 1113857 wn minus zrang

le wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 4τnfn wn( 1113857

wn minus z

2

+ρ2nf

2n wn( 1113857

nablafn wn( 1113857

2 minus

4ρnf2n wn( 1113857

nablafn wn( 1113857

2

wn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

le wn minus z

2

(72)

So combining (71) and (72) we get

xn+1 minus z

βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus z

PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z + βn(u minus z)

le PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z

+ βnu minus z

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 1113857 minus 1 minus βn( 1113857z

+ βnu minus z

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus z

+ βnu minus z

le 1 minus βn( 1113857 wn minus z

+ βnu minus z

le 1 minus βn( 1113857 xn minus z

+ θn xn minus xnminus 1

1113960 1113961 + βnu minus z

le 1 minus βn( 1113857 xn minus z

+ βn σn +u minus z1113858 1113859

(73)

where σn (1 minus βn)(θnβn)xn minus xnminus 1 According to hy-pothesis θn

θn leεn

xn minus xnminus 1

rArrθn

βn

xn minus xnminus 1

leεn

βn

⟶ 0 (74)

Note that

limn⟶infin

σn limn⟶infin

1 minus βn( 1113857θn

βn

xn minus xnminus 1

0 (75)

which implies that the sequence σn1113864 1113865 is bounded Setting

M max supnisinN

σn u minus z1113896 1113897 (76)

as well as using Lemma 7 we conclude that the sequencexn minus z1113864 1113865 is bounded )is shows that the sequence xn1113864 1113865 isbounded and so is wn1113864 1113865

Since xn minus z1113864 1113865 is bounded assume that there exists aconstant M1 such that xn minus zleM1 )us

wn minus z

2 le xn minus z

+ θn xn minus xnminus 1

1113872 1113873

2

xn minus z

2

+ θ2n xn minus xnminus 1

2

+ 2θn xn minus xnminus 1

middot xn minus z

le xn minus z

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

(77)

and we get

8 Journal of Mathematics

xn+1 minus z

2

βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus z

2

PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z + βn(u minus z)

2

le PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z

2

+ 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 1113857 minus 1 minus βn( 1113857z

2

+ 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 wn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

⎡⎢⎢⎣ ⎤⎥⎥⎦ + 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 xn minus z

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

⎡⎢⎢⎣ ⎤⎥⎥⎦ + 2βnlangu minus z xn+1 minus zrang

(78)

From (78)

xn+1 minus z

2 le 1 minus βn( 1113857 xn minus z

2

+ βn

θn

βn

xn minus xnminus 1

2

+ 2M11113890

middotθn

βn

xn minus xnminus 1

+ 2langu minus z xn+1 minus zrang1113891

xn+1 minus z

2 le xn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang(79)

Let

sn xn minus z

2

δn θn

βn

xn minus xnminus 1

2

+ 2M1 middotθn

βn

xn minus xnminus 1

+ 2langu minus z xn+1 minus zrang

ηn ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

cn θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang

(80)

)en (78) can reduce to the inequalities

sn+1 le 1 minus βn( 1113857sn + βnδn nge 1

sn+1 le sn minus ηn + cn(81)

Furthermore we know that

1113944

infin

n0βn infin (82)

limn⟶infin

cn limn⟶infin

θn xn minus xnminus 1

2

+ 2M11113876

middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang1113961 0

(83)

Let nk1113864 1113865 be a subsequence of n and suppose that

limk⟶infin

ηnk 0 (84)

)en we have

limk⟶infin

ρnk4 minus ρnk

1113872 1113873f2nk

wnk1113872 1113873

nablafnkwnk

1113872 1113873

2 0 (85)

which implies by our assumption that

f2nk

wnk1113872 1113873

nablafnkwnk

1113872 1113873

2⟶ 0 as k⟶infin (86)

Since nablafnk(wnk

)1113966 1113967 is bounded it follows thatfnk

(wnk)⟶ 0 as k⟶infin so we get

limk⟶infin(I minus PQnk

)Awnk 0

We next show that ωw(xn) sub S Since xn1113864 1113865 is boundedthe set ωw(xn) is nonempty Let xlowast isin ωw(xn) then thereexists a subsequence xnk

1113966 1113967 of xn1113864 1113865 such that xnkxlowast

wn minus xn

xn + θn xn minus xnminus 1( 1113857 minus xn

θn xn minus xnminus 1

⟶ 0

(87)

and then wnjxlowast and A is a bounded linear operator so

AwnjAxlowast

Since PQnjAwnjisin Qnj

we have

q Awnj1113874 1113875 +langζnj

PQnj

Awnjminus Awnjrang +

β2

PQnj

Awnjminus Awnj

2le 0

(88)

Journal of Mathematics 9

where ζnjisinzq(Awnj

) and by the boundedness of zq we get

q Awnj1113874 1113875le ζnj

middot I minus PQnj1113874 1113875Awnj

minusβ2

I minus PQnj1113874 1113875Awnj

2⟶ 0

(89)

and using the weak lower semicontinuity of q

q Axlowast

( 1113857le lim infj⟶infin

q Awnj1113874 1113875le 0 (90)

)us Axlowast isin QOn the other hand

xn+1 minus wn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus wn

+ βnu minus z

1 minus βn( 1113857 middotρnfn wn( 1113857

nablafn wn( 1113857

+ βnu minus z⟶ 0

(91)

Since (xnj+1 minus βnju) isin Cnj

we have

c wnj1113874 1113875 +langξnj

xnj+1 minus βnju minus wnjrang +

α2

xnj+1 minus βnju minus wnj

2le 0

(92)

where ξnjisinzc(wnj

) and by the boundedness of zc we get

c wnj1113874 1113875le ξnj

middot wnjminus xnj+1 + βnj

u

minusα2

xnj+1 minus wnjminus βnj

u

2

le ξnj

middot wnjminus xnj+1

+ βnju1113876 1113877

minusα2

xnj+1 minus wnj

2

+ β2nju

2minus 2langxnj+1 minus wnj

βnjurang1113876 1113877⟶ 0

(93)

and using the weak lower semicontinuity of c

c xlowast

( 1113857le lim infj⟶infin

c wnj1113874 1113875le 0 (94)

)us xlowast isin C then xlowast isin S that is ωw(xn) sub SNext we have

xn+1 minus xn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus xn

+ βn u minus xn

le 1 minus βn( 1113857 wn minus xn

+ τn nablafn wn( 1113857

1113960 1113961 + βn xn minus u

le wn minus xn

+

ρnfn wn( 1113857

nablafn wn( 1113857

+ βn xn minus u

⟶ 0

(95)

For z PSu and xnkxlowast isin S using Lemma 1

langu minus z xlowast minus zrangle 0 so

lim supn⟶infinlangu minus z xn minus zrang lim sup

k⟶infinlangu minus z xnk

minus zrang

langu minus z xlowast

minus zrang le 0

(96)

and then

lim supn⟶infinlangu minus z xn+1 minus zrang

lim supn⟶infinlangu minus z xn+1 minus xnrang +langu minus z xn minus zrang( 1113857le 0

(97)

and thus

lim supk

δnk lim sup

k

θnk

βnk

xnkminus xnkminus 1

2

+ 2M11113890

middotθnk

βnk

xnkminus xnkminus 1

+ 2langu minus z xnk+1 minus zrang1113891le 0

(98)

From (82) (83) (98) and Lemma 8 we conclude that thesequence xn1113864 1113865 converges strongly to z PSu )e proof iscomplete

4 Numerical Experiments

In this section we present a numerical experiment to il-lustrate the performance of the proposed algorithms Ournumerical experiments are coded in MATLAB R2007running on personal computer with 350GHz Intel Core i3and 4GB RAM In what follows we apply our algorithms tosolve the problem of least absolute shrinkage and selectionoperator which requires solving a convex optimizationproblem as

minxisinRn

12Ax minus y

2

st x1 le t0

(99)

where A isin Rmtimesn y isin Rm and t0 gt 0 are given elements Inour experiment we first generate an m times n matrix A ran-domly by a standardized normal distribution and x is asparse signal with n elements only K of which is nonzerowhich is also generated randomly )e observation y isgenerated as y Ax )e parameters in this experiment areset with n 512 m 256 ε 10minus 4 and t0 K In thissituation it is readily seen that C x isin Rn c(x)le 0 withc(x) x1 minus t0 and Q y1113864 1113865 which in turn implies that

Cn x isin Rn langξn xrang le langξn wnrang minus wn

1 + t01113966 1113967 (100)

where ξn is defined by

ξn( 1113857i

1 if ξn( 1113857igt 0

[minus 1 1] if ξn( 1113857i 0

minus 1 if ξn( 1113857ilt 0

⎧⎪⎪⎨

⎪⎪⎩(101)

standing for the subdifferential of middot 1 As a half-space theassociated projection onto Cn takes the following form

10 Journal of Mathematics

PCn(x)

x +langξn wn minus xrang minus wn

1 + t0

ξn

2 ξn if x notin Cn

x if x isin Cn

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(102)

To show the efficiency of our algorithm we compareit with the algorithm proposed in [40] )e only dif-ference of these two algorithms is that there are no in-ertial terms in the algorithm proposed in [40] For theconvenience we denote Algorithm 1 by Algo I and thealgorithm in [40] by Algo II respectively In Algo-rithm 1 we set

θn

min 081

n2

xn minus xnminus 1

2

⎧⎨

⎩

⎫⎬

⎭ if xn ne xnminus 1

08 if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

τn

1n Alowast

Awn minus y( 1113857

if A

lowastAwn minus y( 1113857

ne 0

0 if Alowast

Awn minus y( 1113857

0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(103)

In Algo II we set θn equiv 0 and τn is chosen the same asabove )e stopping criterion is that xk+1 minus xklt ε )einitial points are x0 (0 0 0)T andx1 100(1 1 1)T )e numerical results of these twoalgorithms with different choices of the sparsity number K

are listed in Figures 1ndash4 It is easy to see that Algo Iconverges faster than Algo II does which indicates that ourmodified algorithm indeed accelerates the convergence ofthe original algorithm

106

104

102

100

||xR

ndash x R

+1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 1 Iterative results with K 50

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 2 Iterative results with K 40

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 3 Iterative results with K 30

104

102

100

||xR

ndash x R

ndash1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 4 Iterative results with K 20

Journal of Mathematics 11

5 Conclusions

In this paper we present two inertial relaxed CQ algorithmsfor solving split feasibility problems in Hilbert spaces byadopting variable step size )ese algorithms adopt the newconvex subset form and it is easy to calculate the projectionsonto these sets Furthermore step size selection in the al-gorithms does not depend on the operator norm )econvergence theorems are established under some mildconditions and a numerical experiment is given to illustratethe performance of the proposed algorithms

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was partially supported by the National NaturalScience Foundation of China (no 11771126)

References

[1] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquo Numerical Algo-rithms vol 8 no 2 pp 221ndash239 1994

[2] C Byrne ldquoA unified treatment of some iterative algorithms insignal processing and image reconstructionrdquo Inverse Prob-lems vol 20 no 1 pp 103ndash120 2004

[3] Y Censor T Elfving N Kopf and T Bortfeld ldquo)emultiple-sets split feasibility problem and its applications for inverseproblemsrdquo Inverse Problems vol 21 no 6 pp 2071ndash20842005

[4] C Byrne ldquoIterative oblique projection onto convex sets andthe split feasibility problemrdquo Inverse Problems vol 18 no 2pp 441ndash453 2002

[5] S He H Tian and H K Xu ldquo)e selective projection methodfor convex feasibility and split feasibility problemsrdquo Journal ofNonlinear and Convex Analysis vol 19 pp 1199ndash1215 2018

[6] S He Z Zhao and B Luo ldquoA relaxed self-adaptive CQ al-gorithm for the multiple-sets split feasibility problemrdquo Op-timization vol 64 no 9 pp 1907ndash1918 2015

[7] G L Acedo V Martin-Marquez F Wang and H XuldquoSolving the split feasibility problem without prior knowledgeof matrix normsrdquo Inverse Problems vol 28 Article ID 0850042012

[8] B Qu and N Xiu ldquoA new halfspace-relaxation projectionmethod for the split feasibility problemrdquo Linear Algebra andits Applications vol 428 no 5-6 pp 1218ndash1229 2008

[9] F Wang ldquoA splitting-relaxed projection method for solvingthe split feasibility problemrdquo Fixed Point gteory vol 14pp 211ndash218 2013

[10] F Wang ldquoOn the convergence of CQ algorithm with variablesteps for the split equality problemrdquo Numerical Algorithmsvol 74 no 3 pp 927ndash935 2017

[11] F Wang ldquoPolyakrsquos gradient method for split feasibilityproblem constrained by level setsrdquo Numerical Algorithmsvol 77 no 3 pp 925ndash938 2018

[12] H-K Xu ldquoA variable Krasnoselrsquoskii-Mann algorithm and themultiple-set split feasibility problemrdquo Inverse Problemsvol 22 no 6 pp 2021ndash2034 2006

[13] Q Yang ldquo)e relaxed CQ algorithm solving the split feasi-bility problemrdquo Inverse Problems vol 20 no 4pp 1261ndash1266 2004

[14] F Alvarez and H Attouch ldquoAn inertial proximal method formaximal monotone operators via discretization of a nonlinearoscillator with dampingrdquo Set-Valued Analysis vol 9 pp 3ndash112001

[15] Y Dang J Sun J Sun and H Xu ldquoInertial accelerated al-gorithms for solving a split feasibility problemrdquo Journal ofIndustrial amp Management Optimization vol 13 no 3pp 1383ndash1394 2017

[16] A Gibali D Mai D )i Mai and N )e Vinh ldquoA newrelaxed CQ algorithm for solving split feasibility problems inHilbert spaces and its applicationsrdquo Journal of Industrial ampManagement Optimization vol 15 no 2 pp 963ndash984 2019

[17] Q L Dong Y J Cho and T M Rassias ldquoGeneral inertialMann algorithms and their convergence analysis for non-expansive mappingsrdquo in Applications of Nonlinear Analysispp 175ndash191 Springer Berlin Germany 2018

[18] Q L Dong Y J Cho L L Zhong and T M Rassias ldquoInertialprojection and contraction algorithms for variational in-equalitiesrdquo Journal of Global Optimization vol 70 no 3pp 687ndash704 2018

[19] Q L Dong J Z Huang X H Li Y J Cho and T M RassiasldquoMiKM multi-step inertial Krasnoselrsquoskiǐ-Mann algorithmand its applicationsrdquo Journal of Global Optimization vol 73no 4 pp 801ndash824 2019

[20] Q L Dong H B Yuan Y J Cho and T M RassiasldquoModified inertial Mann algorithm and inertial CQ-algorithmfor nonexpansive mappingsrdquo Optimization Letters vol 12no 1 pp 87ndash102 2018

[21] Y Shehu and A Gibali ldquoNew inertial relaxed method forsolving split feasibilitiesrdquo Optimization Letters 2020

[22] S S Tai N Pholasa and P Cholamjiak ldquo)emodified inertialrelaxed CQ algorithm for solving the split feasibility prob-lemsrdquo Journal of Industrial and Management Optimizationvol 14 pp 1595ndash1615 2018

[23] S S Tai N Pholasa and P Cholamjiak ldquoRelaxed CQ algo-rithms involving the inertial technique for multiple-sets splitfeasibility problemsrdquo Revista de la Real Academia de CienciasExactas Fisicas y Naturales-Serie A Matematicas vol 113pp 1081ndash1099 2019

[24] S Y Cho ldquoA convergence theorem for generalized mixedequilibrium problems and multivalued asymptotically non-expansive mappingsrdquo Journal of Nonlinear and ConvexAnalysis vol 21 pp 1017ndash1026 2020

[25] H-K Xu ldquoIterative methods for the split feasibility problemin infinite-dimensional Hilbert spacesrdquo Inverse Problemsvol 26 no 10 Article ID 105018 2010

[26] F Wang and H Xu ldquoApproximating curve and strongconvergence of the CQ algorithm for the split feasibilityproblemrdquo Journal of Inequalities and Applications vol 2010Article ID 102085 2010

[27] M Tian and H F Zhang ldquo)e regularized CQ algorithmwithout a priori knowledge of operator norm for solving thesplit feasibility problemrdquo Journal of Inequalities and Appli-cations vol 2017 p 207 2017

[28] Y H Yao W J Gang and Y C Liou ldquoRegularized methodsfor the split feasibility problemrdquo Abstract and AppliedAnalysis vol 2012 Article ID 140679 2012

12 Journal of Mathematics

[29] Y Zhou Z Haiyun and P Wang ldquoIterative methods forfinding the minimum-norm solution of the standardmonotone variational inequality problems with applicationsin Hilbert spacesrdquo Journal of Inequalities and Applicationsvol 2015 2015

[30] T V Nguyen C Prasit and S Suthep ldquoA new CQ algorithmfor solving split feasibility problems in Hilbert spacesrdquoMalaysian Mathematical Sciences Society and Penerbit Uni-versiti Sains Malaysia vol 42 no 5 2020

[31] D R Sahu Y J Cho Q L Dong M R Kashyap and X H LiInertial Relaxed CQ algorithms for Solving a Split FeasibilityProblem in Hilbert Spaces Springer Science Business MediaBerlin Germany 2020

[32] R P Agarwal D O Regan and D R Sahu ldquoFixed pointtheory for Lipschitzian-type mappings with applicationsrdquo inTopological Fixed Point gteory and Its Applications SpringerNew York NY USA 2009

[33] S Y Cho ldquoAmonotone Bregan projection algorithm for fixedpoint and equilibrium problems in a relative Banach spacerdquoFilomat vol 34 pp 1487ndash1497 2020

[34] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator gteory in Hilbert Spaces Springer NewYork NY USA 2011

[35] P E Mainge ldquoInertial iterative process for fixed points ofcertain quasi-nonexpansive mappingrdquo Set Valued Analysisvol 15 pp 67ndash79 2007

[36] P E Mainge ldquoApproximation methods for common fixedpoints of nonexpansive mappings in Hilbert spacesrdquo Journalof Mathematical Analysis and Applications vol 325pp 469ndash479 2007

[37] A Moudafi and A Gibali ldquoregularization of split feasibilityproblemsrdquo Numerical Algorithmsvol 78 no 3 pp 1ndash192017

[38] S He and C Yang ldquoSolving the variational inequality problemdefined on intersection of finite level setsrdquo Abstract andApplied Analysis vol 2013 Article ID 942315 2013

[39] F Wang and H Yu ldquoAn inertial relaxed CQ algorithm withan application to the LASSO and elastic netrdquo Optimizationvol 2020 Article ID 1763989 2020

[40] P K Anh N T Vinh and V T Dung ldquoA new self-adaptiveCQ algorithm with an application to the LASSO problemrdquoJournal of Fixed Point gteory and Applications vol 20 p 1422018

Journal of Mathematics 13

wn minus z

xn + θn xn minus xnminus 1( 1113857 minus z

le xn minus z

+ θn xn minus xnminus 1

(71)

wn minus τnnablafn wn( 1113857 minus z

2

wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 2τnlangnablafn wn( 1113857 wn minus zrang

le wn minus z

2

+ τ2n nablafn wn( 1113857

2

minus 4τnfn wn( 1113857

wn minus z

2

+ρ2nf

2n wn( 1113857

nablafn wn( 1113857

2 minus

4ρnf2n wn( 1113857

nablafn wn( 1113857

2

wn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

le wn minus z

2

(72)

So combining (71) and (72) we get

xn+1 minus z

βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus z

PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z + βn(u minus z)

le PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z

+ βnu minus z

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 1113857 minus 1 minus βn( 1113857z

+ βnu minus z

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus z

+ βnu minus z

le 1 minus βn( 1113857 wn minus z

+ βnu minus z

le 1 minus βn( 1113857 xn minus z

+ θn xn minus xnminus 1

1113960 1113961 + βnu minus z

le 1 minus βn( 1113857 xn minus z

+ βn σn +u minus z1113858 1113859

(73)

where σn (1 minus βn)(θnβn)xn minus xnminus 1 According to hy-pothesis θn

θn leεn

xn minus xnminus 1

rArrθn

βn

xn minus xnminus 1

leεn

βn

⟶ 0 (74)

Note that

limn⟶infin

σn limn⟶infin

1 minus βn( 1113857θn

βn

xn minus xnminus 1

0 (75)

which implies that the sequence σn1113864 1113865 is bounded Setting

M max supnisinN

σn u minus z1113896 1113897 (76)

as well as using Lemma 7 we conclude that the sequencexn minus z1113864 1113865 is bounded )is shows that the sequence xn1113864 1113865 isbounded and so is wn1113864 1113865

Since xn minus z1113864 1113865 is bounded assume that there exists aconstant M1 such that xn minus zleM1 )us

wn minus z

2 le xn minus z

+ θn xn minus xnminus 1

1113872 1113873

2

xn minus z

2

+ θ2n xn minus xnminus 1

2

+ 2θn xn minus xnminus 1

middot xn minus z

le xn minus z

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

(77)

and we get

8 Journal of Mathematics

xn+1 minus z

2

βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus z

2

PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z + βn(u minus z)

2

le PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z

2

+ 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 1113857 minus 1 minus βn( 1113857z

2

+ 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 wn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

⎡⎢⎢⎣ ⎤⎥⎥⎦ + 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 xn minus z

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

⎡⎢⎢⎣ ⎤⎥⎥⎦ + 2βnlangu minus z xn+1 minus zrang

(78)

From (78)

xn+1 minus z

2 le 1 minus βn( 1113857 xn minus z

2

+ βn

θn

βn

xn minus xnminus 1

2

+ 2M11113890

middotθn

βn

xn minus xnminus 1

+ 2langu minus z xn+1 minus zrang1113891

xn+1 minus z

2 le xn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang(79)

Let

sn xn minus z

2

δn θn

βn

xn minus xnminus 1

2

+ 2M1 middotθn

βn

xn minus xnminus 1

+ 2langu minus z xn+1 minus zrang

ηn ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

cn θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang

(80)

)en (78) can reduce to the inequalities

sn+1 le 1 minus βn( 1113857sn + βnδn nge 1

sn+1 le sn minus ηn + cn(81)

Furthermore we know that

1113944

infin

n0βn infin (82)

limn⟶infin

cn limn⟶infin

θn xn minus xnminus 1

2

+ 2M11113876

middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang1113961 0

(83)

Let nk1113864 1113865 be a subsequence of n and suppose that

limk⟶infin

ηnk 0 (84)

)en we have

limk⟶infin

ρnk4 minus ρnk

1113872 1113873f2nk

wnk1113872 1113873

nablafnkwnk

1113872 1113873

2 0 (85)

which implies by our assumption that

f2nk

wnk1113872 1113873

nablafnkwnk

1113872 1113873

2⟶ 0 as k⟶infin (86)

Since nablafnk(wnk

)1113966 1113967 is bounded it follows thatfnk

(wnk)⟶ 0 as k⟶infin so we get

limk⟶infin(I minus PQnk

)Awnk 0

We next show that ωw(xn) sub S Since xn1113864 1113865 is boundedthe set ωw(xn) is nonempty Let xlowast isin ωw(xn) then thereexists a subsequence xnk

1113966 1113967 of xn1113864 1113865 such that xnkxlowast

wn minus xn

xn + θn xn minus xnminus 1( 1113857 minus xn

θn xn minus xnminus 1

⟶ 0

(87)

and then wnjxlowast and A is a bounded linear operator so

AwnjAxlowast

Since PQnjAwnjisin Qnj

we have

q Awnj1113874 1113875 +langζnj

PQnj

Awnjminus Awnjrang +

β2

PQnj

Awnjminus Awnj

2le 0

(88)

Journal of Mathematics 9

where ζnjisinzq(Awnj

) and by the boundedness of zq we get

q Awnj1113874 1113875le ζnj

middot I minus PQnj1113874 1113875Awnj

minusβ2

I minus PQnj1113874 1113875Awnj

2⟶ 0

(89)

and using the weak lower semicontinuity of q

q Axlowast

( 1113857le lim infj⟶infin

q Awnj1113874 1113875le 0 (90)

)us Axlowast isin QOn the other hand

xn+1 minus wn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus wn

+ βnu minus z

1 minus βn( 1113857 middotρnfn wn( 1113857

nablafn wn( 1113857

+ βnu minus z⟶ 0

(91)

Since (xnj+1 minus βnju) isin Cnj

we have

c wnj1113874 1113875 +langξnj

xnj+1 minus βnju minus wnjrang +

α2

xnj+1 minus βnju minus wnj

2le 0

(92)

where ξnjisinzc(wnj

) and by the boundedness of zc we get

c wnj1113874 1113875le ξnj

middot wnjminus xnj+1 + βnj

u

minusα2

xnj+1 minus wnjminus βnj

u

2

le ξnj

middot wnjminus xnj+1

+ βnju1113876 1113877

minusα2

xnj+1 minus wnj

2

+ β2nju

2minus 2langxnj+1 minus wnj

βnjurang1113876 1113877⟶ 0

(93)

and using the weak lower semicontinuity of c

c xlowast

( 1113857le lim infj⟶infin

c wnj1113874 1113875le 0 (94)

)us xlowast isin C then xlowast isin S that is ωw(xn) sub SNext we have

xn+1 minus xn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus xn

+ βn u minus xn

le 1 minus βn( 1113857 wn minus xn

+ τn nablafn wn( 1113857

1113960 1113961 + βn xn minus u

le wn minus xn

+

ρnfn wn( 1113857

nablafn wn( 1113857

+ βn xn minus u

⟶ 0

(95)

For z PSu and xnkxlowast isin S using Lemma 1

langu minus z xlowast minus zrangle 0 so

lim supn⟶infinlangu minus z xn minus zrang lim sup

k⟶infinlangu minus z xnk

minus zrang

langu minus z xlowast

minus zrang le 0

(96)

and then

lim supn⟶infinlangu minus z xn+1 minus zrang

lim supn⟶infinlangu minus z xn+1 minus xnrang +langu minus z xn minus zrang( 1113857le 0

(97)

and thus

lim supk

δnk lim sup

k

θnk

βnk

xnkminus xnkminus 1

2

+ 2M11113890

middotθnk

βnk

xnkminus xnkminus 1

+ 2langu minus z xnk+1 minus zrang1113891le 0

(98)

From (82) (83) (98) and Lemma 8 we conclude that thesequence xn1113864 1113865 converges strongly to z PSu )e proof iscomplete

4 Numerical Experiments

In this section we present a numerical experiment to il-lustrate the performance of the proposed algorithms Ournumerical experiments are coded in MATLAB R2007running on personal computer with 350GHz Intel Core i3and 4GB RAM In what follows we apply our algorithms tosolve the problem of least absolute shrinkage and selectionoperator which requires solving a convex optimizationproblem as

minxisinRn

12Ax minus y

2

st x1 le t0

(99)

where A isin Rmtimesn y isin Rm and t0 gt 0 are given elements Inour experiment we first generate an m times n matrix A ran-domly by a standardized normal distribution and x is asparse signal with n elements only K of which is nonzerowhich is also generated randomly )e observation y isgenerated as y Ax )e parameters in this experiment areset with n 512 m 256 ε 10minus 4 and t0 K In thissituation it is readily seen that C x isin Rn c(x)le 0 withc(x) x1 minus t0 and Q y1113864 1113865 which in turn implies that

Cn x isin Rn langξn xrang le langξn wnrang minus wn

1 + t01113966 1113967 (100)

where ξn is defined by

ξn( 1113857i

1 if ξn( 1113857igt 0

[minus 1 1] if ξn( 1113857i 0

minus 1 if ξn( 1113857ilt 0

⎧⎪⎪⎨

⎪⎪⎩(101)

standing for the subdifferential of middot 1 As a half-space theassociated projection onto Cn takes the following form

10 Journal of Mathematics

PCn(x)

x +langξn wn minus xrang minus wn

1 + t0

ξn

2 ξn if x notin Cn

x if x isin Cn

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(102)

To show the efficiency of our algorithm we compareit with the algorithm proposed in [40] )e only dif-ference of these two algorithms is that there are no in-ertial terms in the algorithm proposed in [40] For theconvenience we denote Algorithm 1 by Algo I and thealgorithm in [40] by Algo II respectively In Algo-rithm 1 we set

θn

min 081

n2

xn minus xnminus 1

2

⎧⎨

⎩

⎫⎬

⎭ if xn ne xnminus 1

08 if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

τn

1n Alowast

Awn minus y( 1113857

if A

lowastAwn minus y( 1113857

ne 0

0 if Alowast

Awn minus y( 1113857

0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(103)

In Algo II we set θn equiv 0 and τn is chosen the same asabove )e stopping criterion is that xk+1 minus xklt ε )einitial points are x0 (0 0 0)T andx1 100(1 1 1)T )e numerical results of these twoalgorithms with different choices of the sparsity number K

are listed in Figures 1ndash4 It is easy to see that Algo Iconverges faster than Algo II does which indicates that ourmodified algorithm indeed accelerates the convergence ofthe original algorithm

106

104

102

100

||xR

ndash x R

+1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 1 Iterative results with K 50

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 2 Iterative results with K 40

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 3 Iterative results with K 30

104

102

100

||xR

ndash x R

ndash1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 4 Iterative results with K 20

Journal of Mathematics 11

5 Conclusions

In this paper we present two inertial relaxed CQ algorithmsfor solving split feasibility problems in Hilbert spaces byadopting variable step size )ese algorithms adopt the newconvex subset form and it is easy to calculate the projectionsonto these sets Furthermore step size selection in the al-gorithms does not depend on the operator norm )econvergence theorems are established under some mildconditions and a numerical experiment is given to illustratethe performance of the proposed algorithms

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was partially supported by the National NaturalScience Foundation of China (no 11771126)

References

[1] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquo Numerical Algo-rithms vol 8 no 2 pp 221ndash239 1994

[2] C Byrne ldquoA unified treatment of some iterative algorithms insignal processing and image reconstructionrdquo Inverse Prob-lems vol 20 no 1 pp 103ndash120 2004

[3] Y Censor T Elfving N Kopf and T Bortfeld ldquo)emultiple-sets split feasibility problem and its applications for inverseproblemsrdquo Inverse Problems vol 21 no 6 pp 2071ndash20842005

[4] C Byrne ldquoIterative oblique projection onto convex sets andthe split feasibility problemrdquo Inverse Problems vol 18 no 2pp 441ndash453 2002

[5] S He H Tian and H K Xu ldquo)e selective projection methodfor convex feasibility and split feasibility problemsrdquo Journal ofNonlinear and Convex Analysis vol 19 pp 1199ndash1215 2018

[6] S He Z Zhao and B Luo ldquoA relaxed self-adaptive CQ al-gorithm for the multiple-sets split feasibility problemrdquo Op-timization vol 64 no 9 pp 1907ndash1918 2015

[7] G L Acedo V Martin-Marquez F Wang and H XuldquoSolving the split feasibility problem without prior knowledgeof matrix normsrdquo Inverse Problems vol 28 Article ID 0850042012

[8] B Qu and N Xiu ldquoA new halfspace-relaxation projectionmethod for the split feasibility problemrdquo Linear Algebra andits Applications vol 428 no 5-6 pp 1218ndash1229 2008

[9] F Wang ldquoA splitting-relaxed projection method for solvingthe split feasibility problemrdquo Fixed Point gteory vol 14pp 211ndash218 2013

[10] F Wang ldquoOn the convergence of CQ algorithm with variablesteps for the split equality problemrdquo Numerical Algorithmsvol 74 no 3 pp 927ndash935 2017

[11] F Wang ldquoPolyakrsquos gradient method for split feasibilityproblem constrained by level setsrdquo Numerical Algorithmsvol 77 no 3 pp 925ndash938 2018

[12] H-K Xu ldquoA variable Krasnoselrsquoskii-Mann algorithm and themultiple-set split feasibility problemrdquo Inverse Problemsvol 22 no 6 pp 2021ndash2034 2006

[13] Q Yang ldquo)e relaxed CQ algorithm solving the split feasi-bility problemrdquo Inverse Problems vol 20 no 4pp 1261ndash1266 2004

[14] F Alvarez and H Attouch ldquoAn inertial proximal method formaximal monotone operators via discretization of a nonlinearoscillator with dampingrdquo Set-Valued Analysis vol 9 pp 3ndash112001

[15] Y Dang J Sun J Sun and H Xu ldquoInertial accelerated al-gorithms for solving a split feasibility problemrdquo Journal ofIndustrial amp Management Optimization vol 13 no 3pp 1383ndash1394 2017

[16] A Gibali D Mai D )i Mai and N )e Vinh ldquoA newrelaxed CQ algorithm for solving split feasibility problems inHilbert spaces and its applicationsrdquo Journal of Industrial ampManagement Optimization vol 15 no 2 pp 963ndash984 2019

[17] Q L Dong Y J Cho and T M Rassias ldquoGeneral inertialMann algorithms and their convergence analysis for non-expansive mappingsrdquo in Applications of Nonlinear Analysispp 175ndash191 Springer Berlin Germany 2018

[18] Q L Dong Y J Cho L L Zhong and T M Rassias ldquoInertialprojection and contraction algorithms for variational in-equalitiesrdquo Journal of Global Optimization vol 70 no 3pp 687ndash704 2018

[19] Q L Dong J Z Huang X H Li Y J Cho and T M RassiasldquoMiKM multi-step inertial Krasnoselrsquoskiǐ-Mann algorithmand its applicationsrdquo Journal of Global Optimization vol 73no 4 pp 801ndash824 2019

[20] Q L Dong H B Yuan Y J Cho and T M RassiasldquoModified inertial Mann algorithm and inertial CQ-algorithmfor nonexpansive mappingsrdquo Optimization Letters vol 12no 1 pp 87ndash102 2018

[21] Y Shehu and A Gibali ldquoNew inertial relaxed method forsolving split feasibilitiesrdquo Optimization Letters 2020

[22] S S Tai N Pholasa and P Cholamjiak ldquo)emodified inertialrelaxed CQ algorithm for solving the split feasibility prob-lemsrdquo Journal of Industrial and Management Optimizationvol 14 pp 1595ndash1615 2018

[23] S S Tai N Pholasa and P Cholamjiak ldquoRelaxed CQ algo-rithms involving the inertial technique for multiple-sets splitfeasibility problemsrdquo Revista de la Real Academia de CienciasExactas Fisicas y Naturales-Serie A Matematicas vol 113pp 1081ndash1099 2019

[24] S Y Cho ldquoA convergence theorem for generalized mixedequilibrium problems and multivalued asymptotically non-expansive mappingsrdquo Journal of Nonlinear and ConvexAnalysis vol 21 pp 1017ndash1026 2020

[25] H-K Xu ldquoIterative methods for the split feasibility problemin infinite-dimensional Hilbert spacesrdquo Inverse Problemsvol 26 no 10 Article ID 105018 2010

[26] F Wang and H Xu ldquoApproximating curve and strongconvergence of the CQ algorithm for the split feasibilityproblemrdquo Journal of Inequalities and Applications vol 2010Article ID 102085 2010

[27] M Tian and H F Zhang ldquo)e regularized CQ algorithmwithout a priori knowledge of operator norm for solving thesplit feasibility problemrdquo Journal of Inequalities and Appli-cations vol 2017 p 207 2017

[28] Y H Yao W J Gang and Y C Liou ldquoRegularized methodsfor the split feasibility problemrdquo Abstract and AppliedAnalysis vol 2012 Article ID 140679 2012

12 Journal of Mathematics

[29] Y Zhou Z Haiyun and P Wang ldquoIterative methods forfinding the minimum-norm solution of the standardmonotone variational inequality problems with applicationsin Hilbert spacesrdquo Journal of Inequalities and Applicationsvol 2015 2015

[30] T V Nguyen C Prasit and S Suthep ldquoA new CQ algorithmfor solving split feasibility problems in Hilbert spacesrdquoMalaysian Mathematical Sciences Society and Penerbit Uni-versiti Sains Malaysia vol 42 no 5 2020

[31] D R Sahu Y J Cho Q L Dong M R Kashyap and X H LiInertial Relaxed CQ algorithms for Solving a Split FeasibilityProblem in Hilbert Spaces Springer Science Business MediaBerlin Germany 2020

[32] R P Agarwal D O Regan and D R Sahu ldquoFixed pointtheory for Lipschitzian-type mappings with applicationsrdquo inTopological Fixed Point gteory and Its Applications SpringerNew York NY USA 2009

[33] S Y Cho ldquoAmonotone Bregan projection algorithm for fixedpoint and equilibrium problems in a relative Banach spacerdquoFilomat vol 34 pp 1487ndash1497 2020

[34] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator gteory in Hilbert Spaces Springer NewYork NY USA 2011

[35] P E Mainge ldquoInertial iterative process for fixed points ofcertain quasi-nonexpansive mappingrdquo Set Valued Analysisvol 15 pp 67ndash79 2007

[36] P E Mainge ldquoApproximation methods for common fixedpoints of nonexpansive mappings in Hilbert spacesrdquo Journalof Mathematical Analysis and Applications vol 325pp 469ndash479 2007

[37] A Moudafi and A Gibali ldquoregularization of split feasibilityproblemsrdquo Numerical Algorithmsvol 78 no 3 pp 1ndash192017

[38] S He and C Yang ldquoSolving the variational inequality problemdefined on intersection of finite level setsrdquo Abstract andApplied Analysis vol 2013 Article ID 942315 2013

[39] F Wang and H Yu ldquoAn inertial relaxed CQ algorithm withan application to the LASSO and elastic netrdquo Optimizationvol 2020 Article ID 1763989 2020

[40] P K Anh N T Vinh and V T Dung ldquoA new self-adaptiveCQ algorithm with an application to the LASSO problemrdquoJournal of Fixed Point gteory and Applications vol 20 p 1422018

Journal of Mathematics 13

xn+1 minus z

2

βnu + PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus z

2

PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z + βn(u minus z)

2

le PCn1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 11138571113858 1113859 minus 1 minus βn( 1113857z

2

+ 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857( 1113857 minus 1 minus βn( 1113857z

2

+ 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 wn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

⎡⎢⎢⎣ ⎤⎥⎥⎦ + 2βnlangu minus z xn+1 minus zrang

le 1 minus βn( 1113857 xn minus z

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

⎡⎢⎢⎣ ⎤⎥⎥⎦ + 2βnlangu minus z xn+1 minus zrang

(78)

From (78)

xn+1 minus z

2 le 1 minus βn( 1113857 xn minus z

2

+ βn

θn

βn

xn minus xnminus 1

2

+ 2M11113890

middotθn

βn

xn minus xnminus 1

+ 2langu minus z xn+1 minus zrang1113891

xn+1 minus z

2 le xn minus z

2

minus ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

+ θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang(79)

Let

sn xn minus z

2

δn θn

βn

xn minus xnminus 1

2

+ 2M1 middotθn

βn

xn minus xnminus 1

+ 2langu minus z xn+1 minus zrang

ηn ρn 4 minus ρn( 1113857f2n wn( 1113857

nablafn wn( 1113857

2

cn θn xn minus xnminus 1

2

+ 2M1 middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang

(80)

)en (78) can reduce to the inequalities

sn+1 le 1 minus βn( 1113857sn + βnδn nge 1

sn+1 le sn minus ηn + cn(81)

Furthermore we know that

1113944

infin

n0βn infin (82)

limn⟶infin

cn limn⟶infin

θn xn minus xnminus 1

2

+ 2M11113876

middot θn xn minus xnminus 1

+ 2βnlangu minus z xn+1 minus zrang1113961 0

(83)

Let nk1113864 1113865 be a subsequence of n and suppose that

limk⟶infin

ηnk 0 (84)

)en we have

limk⟶infin

ρnk4 minus ρnk

1113872 1113873f2nk

wnk1113872 1113873

nablafnkwnk

1113872 1113873

2 0 (85)

which implies by our assumption that

f2nk

wnk1113872 1113873

nablafnkwnk

1113872 1113873

2⟶ 0 as k⟶infin (86)

Since nablafnk(wnk

)1113966 1113967 is bounded it follows thatfnk

(wnk)⟶ 0 as k⟶infin so we get

limk⟶infin(I minus PQnk

)Awnk 0

We next show that ωw(xn) sub S Since xn1113864 1113865 is boundedthe set ωw(xn) is nonempty Let xlowast isin ωw(xn) then thereexists a subsequence xnk

1113966 1113967 of xn1113864 1113865 such that xnkxlowast

wn minus xn

xn + θn xn minus xnminus 1( 1113857 minus xn

θn xn minus xnminus 1

⟶ 0

(87)

and then wnjxlowast and A is a bounded linear operator so

AwnjAxlowast

Since PQnjAwnjisin Qnj

we have

q Awnj1113874 1113875 +langζnj

PQnj

Awnjminus Awnjrang +

β2

PQnj

Awnjminus Awnj

2le 0

(88)

Journal of Mathematics 9

where ζnjisinzq(Awnj

) and by the boundedness of zq we get

q Awnj1113874 1113875le ζnj

middot I minus PQnj1113874 1113875Awnj

minusβ2

I minus PQnj1113874 1113875Awnj

2⟶ 0

(89)

and using the weak lower semicontinuity of q

q Axlowast

( 1113857le lim infj⟶infin

q Awnj1113874 1113875le 0 (90)

)us Axlowast isin QOn the other hand

xn+1 minus wn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus wn

+ βnu minus z

1 minus βn( 1113857 middotρnfn wn( 1113857

nablafn wn( 1113857

+ βnu minus z⟶ 0

(91)

Since (xnj+1 minus βnju) isin Cnj

we have

c wnj1113874 1113875 +langξnj

xnj+1 minus βnju minus wnjrang +

α2

xnj+1 minus βnju minus wnj

2le 0

(92)

where ξnjisinzc(wnj

) and by the boundedness of zc we get

c wnj1113874 1113875le ξnj

middot wnjminus xnj+1 + βnj

u

minusα2

xnj+1 minus wnjminus βnj

u

2

le ξnj

middot wnjminus xnj+1

+ βnju1113876 1113877

minusα2

xnj+1 minus wnj

2

+ β2nju

2minus 2langxnj+1 minus wnj

βnjurang1113876 1113877⟶ 0

(93)

and using the weak lower semicontinuity of c

c xlowast

( 1113857le lim infj⟶infin

c wnj1113874 1113875le 0 (94)

)us xlowast isin C then xlowast isin S that is ωw(xn) sub SNext we have

xn+1 minus xn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus xn

+ βn u minus xn

le 1 minus βn( 1113857 wn minus xn

+ τn nablafn wn( 1113857

1113960 1113961 + βn xn minus u

le wn minus xn

+

ρnfn wn( 1113857

nablafn wn( 1113857

+ βn xn minus u

⟶ 0

(95)

For z PSu and xnkxlowast isin S using Lemma 1

langu minus z xlowast minus zrangle 0 so

lim supn⟶infinlangu minus z xn minus zrang lim sup

k⟶infinlangu minus z xnk

minus zrang

langu minus z xlowast

minus zrang le 0

(96)

and then

lim supn⟶infinlangu minus z xn+1 minus zrang

lim supn⟶infinlangu minus z xn+1 minus xnrang +langu minus z xn minus zrang( 1113857le 0

(97)

and thus

lim supk

δnk lim sup

k

θnk

βnk

xnkminus xnkminus 1

2

+ 2M11113890

middotθnk

βnk

xnkminus xnkminus 1

+ 2langu minus z xnk+1 minus zrang1113891le 0

(98)

From (82) (83) (98) and Lemma 8 we conclude that thesequence xn1113864 1113865 converges strongly to z PSu )e proof iscomplete

4 Numerical Experiments

In this section we present a numerical experiment to il-lustrate the performance of the proposed algorithms Ournumerical experiments are coded in MATLAB R2007running on personal computer with 350GHz Intel Core i3and 4GB RAM In what follows we apply our algorithms tosolve the problem of least absolute shrinkage and selectionoperator which requires solving a convex optimizationproblem as

minxisinRn

12Ax minus y

2

st x1 le t0

(99)

where A isin Rmtimesn y isin Rm and t0 gt 0 are given elements Inour experiment we first generate an m times n matrix A ran-domly by a standardized normal distribution and x is asparse signal with n elements only K of which is nonzerowhich is also generated randomly )e observation y isgenerated as y Ax )e parameters in this experiment areset with n 512 m 256 ε 10minus 4 and t0 K In thissituation it is readily seen that C x isin Rn c(x)le 0 withc(x) x1 minus t0 and Q y1113864 1113865 which in turn implies that

Cn x isin Rn langξn xrang le langξn wnrang minus wn

1 + t01113966 1113967 (100)

where ξn is defined by

ξn( 1113857i

1 if ξn( 1113857igt 0

[minus 1 1] if ξn( 1113857i 0

minus 1 if ξn( 1113857ilt 0

⎧⎪⎪⎨

⎪⎪⎩(101)

standing for the subdifferential of middot 1 As a half-space theassociated projection onto Cn takes the following form

10 Journal of Mathematics

PCn(x)

x +langξn wn minus xrang minus wn

1 + t0

ξn

2 ξn if x notin Cn

x if x isin Cn

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(102)

To show the efficiency of our algorithm we compareit with the algorithm proposed in [40] )e only dif-ference of these two algorithms is that there are no in-ertial terms in the algorithm proposed in [40] For theconvenience we denote Algorithm 1 by Algo I and thealgorithm in [40] by Algo II respectively In Algo-rithm 1 we set

θn

min 081

n2

xn minus xnminus 1

2

⎧⎨

⎩

⎫⎬

⎭ if xn ne xnminus 1

08 if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

τn

1n Alowast

Awn minus y( 1113857

if A

lowastAwn minus y( 1113857

ne 0

0 if Alowast

Awn minus y( 1113857

0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(103)

In Algo II we set θn equiv 0 and τn is chosen the same asabove )e stopping criterion is that xk+1 minus xklt ε )einitial points are x0 (0 0 0)T andx1 100(1 1 1)T )e numerical results of these twoalgorithms with different choices of the sparsity number K

are listed in Figures 1ndash4 It is easy to see that Algo Iconverges faster than Algo II does which indicates that ourmodified algorithm indeed accelerates the convergence ofthe original algorithm

106

104

102

100

||xR

ndash x R

+1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 1 Iterative results with K 50

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 2 Iterative results with K 40

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 3 Iterative results with K 30

104

102

100

||xR

ndash x R

ndash1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 4 Iterative results with K 20

Journal of Mathematics 11

5 Conclusions

In this paper we present two inertial relaxed CQ algorithmsfor solving split feasibility problems in Hilbert spaces byadopting variable step size )ese algorithms adopt the newconvex subset form and it is easy to calculate the projectionsonto these sets Furthermore step size selection in the al-gorithms does not depend on the operator norm )econvergence theorems are established under some mildconditions and a numerical experiment is given to illustratethe performance of the proposed algorithms

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was partially supported by the National NaturalScience Foundation of China (no 11771126)

References

[1] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquo Numerical Algo-rithms vol 8 no 2 pp 221ndash239 1994

[2] C Byrne ldquoA unified treatment of some iterative algorithms insignal processing and image reconstructionrdquo Inverse Prob-lems vol 20 no 1 pp 103ndash120 2004

[3] Y Censor T Elfving N Kopf and T Bortfeld ldquo)emultiple-sets split feasibility problem and its applications for inverseproblemsrdquo Inverse Problems vol 21 no 6 pp 2071ndash20842005

[4] C Byrne ldquoIterative oblique projection onto convex sets andthe split feasibility problemrdquo Inverse Problems vol 18 no 2pp 441ndash453 2002

[5] S He H Tian and H K Xu ldquo)e selective projection methodfor convex feasibility and split feasibility problemsrdquo Journal ofNonlinear and Convex Analysis vol 19 pp 1199ndash1215 2018

[6] S He Z Zhao and B Luo ldquoA relaxed self-adaptive CQ al-gorithm for the multiple-sets split feasibility problemrdquo Op-timization vol 64 no 9 pp 1907ndash1918 2015

[7] G L Acedo V Martin-Marquez F Wang and H XuldquoSolving the split feasibility problem without prior knowledgeof matrix normsrdquo Inverse Problems vol 28 Article ID 0850042012

[8] B Qu and N Xiu ldquoA new halfspace-relaxation projectionmethod for the split feasibility problemrdquo Linear Algebra andits Applications vol 428 no 5-6 pp 1218ndash1229 2008

[9] F Wang ldquoA splitting-relaxed projection method for solvingthe split feasibility problemrdquo Fixed Point gteory vol 14pp 211ndash218 2013

[10] F Wang ldquoOn the convergence of CQ algorithm with variablesteps for the split equality problemrdquo Numerical Algorithmsvol 74 no 3 pp 927ndash935 2017

[11] F Wang ldquoPolyakrsquos gradient method for split feasibilityproblem constrained by level setsrdquo Numerical Algorithmsvol 77 no 3 pp 925ndash938 2018

[12] H-K Xu ldquoA variable Krasnoselrsquoskii-Mann algorithm and themultiple-set split feasibility problemrdquo Inverse Problemsvol 22 no 6 pp 2021ndash2034 2006

[13] Q Yang ldquo)e relaxed CQ algorithm solving the split feasi-bility problemrdquo Inverse Problems vol 20 no 4pp 1261ndash1266 2004

[14] F Alvarez and H Attouch ldquoAn inertial proximal method formaximal monotone operators via discretization of a nonlinearoscillator with dampingrdquo Set-Valued Analysis vol 9 pp 3ndash112001

[15] Y Dang J Sun J Sun and H Xu ldquoInertial accelerated al-gorithms for solving a split feasibility problemrdquo Journal ofIndustrial amp Management Optimization vol 13 no 3pp 1383ndash1394 2017

[16] A Gibali D Mai D )i Mai and N )e Vinh ldquoA newrelaxed CQ algorithm for solving split feasibility problems inHilbert spaces and its applicationsrdquo Journal of Industrial ampManagement Optimization vol 15 no 2 pp 963ndash984 2019

[17] Q L Dong Y J Cho and T M Rassias ldquoGeneral inertialMann algorithms and their convergence analysis for non-expansive mappingsrdquo in Applications of Nonlinear Analysispp 175ndash191 Springer Berlin Germany 2018

[18] Q L Dong Y J Cho L L Zhong and T M Rassias ldquoInertialprojection and contraction algorithms for variational in-equalitiesrdquo Journal of Global Optimization vol 70 no 3pp 687ndash704 2018

[19] Q L Dong J Z Huang X H Li Y J Cho and T M RassiasldquoMiKM multi-step inertial Krasnoselrsquoskiǐ-Mann algorithmand its applicationsrdquo Journal of Global Optimization vol 73no 4 pp 801ndash824 2019

[20] Q L Dong H B Yuan Y J Cho and T M RassiasldquoModified inertial Mann algorithm and inertial CQ-algorithmfor nonexpansive mappingsrdquo Optimization Letters vol 12no 1 pp 87ndash102 2018

[21] Y Shehu and A Gibali ldquoNew inertial relaxed method forsolving split feasibilitiesrdquo Optimization Letters 2020

[22] S S Tai N Pholasa and P Cholamjiak ldquo)emodified inertialrelaxed CQ algorithm for solving the split feasibility prob-lemsrdquo Journal of Industrial and Management Optimizationvol 14 pp 1595ndash1615 2018

[23] S S Tai N Pholasa and P Cholamjiak ldquoRelaxed CQ algo-rithms involving the inertial technique for multiple-sets splitfeasibility problemsrdquo Revista de la Real Academia de CienciasExactas Fisicas y Naturales-Serie A Matematicas vol 113pp 1081ndash1099 2019

[24] S Y Cho ldquoA convergence theorem for generalized mixedequilibrium problems and multivalued asymptotically non-expansive mappingsrdquo Journal of Nonlinear and ConvexAnalysis vol 21 pp 1017ndash1026 2020

[25] H-K Xu ldquoIterative methods for the split feasibility problemin infinite-dimensional Hilbert spacesrdquo Inverse Problemsvol 26 no 10 Article ID 105018 2010

[26] F Wang and H Xu ldquoApproximating curve and strongconvergence of the CQ algorithm for the split feasibilityproblemrdquo Journal of Inequalities and Applications vol 2010Article ID 102085 2010

[27] M Tian and H F Zhang ldquo)e regularized CQ algorithmwithout a priori knowledge of operator norm for solving thesplit feasibility problemrdquo Journal of Inequalities and Appli-cations vol 2017 p 207 2017

[28] Y H Yao W J Gang and Y C Liou ldquoRegularized methodsfor the split feasibility problemrdquo Abstract and AppliedAnalysis vol 2012 Article ID 140679 2012

12 Journal of Mathematics

[29] Y Zhou Z Haiyun and P Wang ldquoIterative methods forfinding the minimum-norm solution of the standardmonotone variational inequality problems with applicationsin Hilbert spacesrdquo Journal of Inequalities and Applicationsvol 2015 2015

[30] T V Nguyen C Prasit and S Suthep ldquoA new CQ algorithmfor solving split feasibility problems in Hilbert spacesrdquoMalaysian Mathematical Sciences Society and Penerbit Uni-versiti Sains Malaysia vol 42 no 5 2020

[31] D R Sahu Y J Cho Q L Dong M R Kashyap and X H LiInertial Relaxed CQ algorithms for Solving a Split FeasibilityProblem in Hilbert Spaces Springer Science Business MediaBerlin Germany 2020

[32] R P Agarwal D O Regan and D R Sahu ldquoFixed pointtheory for Lipschitzian-type mappings with applicationsrdquo inTopological Fixed Point gteory and Its Applications SpringerNew York NY USA 2009

[33] S Y Cho ldquoAmonotone Bregan projection algorithm for fixedpoint and equilibrium problems in a relative Banach spacerdquoFilomat vol 34 pp 1487ndash1497 2020

[34] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator gteory in Hilbert Spaces Springer NewYork NY USA 2011

[35] P E Mainge ldquoInertial iterative process for fixed points ofcertain quasi-nonexpansive mappingrdquo Set Valued Analysisvol 15 pp 67ndash79 2007

[36] P E Mainge ldquoApproximation methods for common fixedpoints of nonexpansive mappings in Hilbert spacesrdquo Journalof Mathematical Analysis and Applications vol 325pp 469ndash479 2007

[37] A Moudafi and A Gibali ldquoregularization of split feasibilityproblemsrdquo Numerical Algorithmsvol 78 no 3 pp 1ndash192017

[38] S He and C Yang ldquoSolving the variational inequality problemdefined on intersection of finite level setsrdquo Abstract andApplied Analysis vol 2013 Article ID 942315 2013

[39] F Wang and H Yu ldquoAn inertial relaxed CQ algorithm withan application to the LASSO and elastic netrdquo Optimizationvol 2020 Article ID 1763989 2020

[40] P K Anh N T Vinh and V T Dung ldquoA new self-adaptiveCQ algorithm with an application to the LASSO problemrdquoJournal of Fixed Point gteory and Applications vol 20 p 1422018

Journal of Mathematics 13

where ζnjisinzq(Awnj

) and by the boundedness of zq we get

q Awnj1113874 1113875le ζnj

middot I minus PQnj1113874 1113875Awnj

minusβ2

I minus PQnj1113874 1113875Awnj

2⟶ 0

(89)

and using the weak lower semicontinuity of q

q Axlowast

( 1113857le lim infj⟶infin

q Awnj1113874 1113875le 0 (90)

)us Axlowast isin QOn the other hand

xn+1 minus wn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus wn

+ βnu minus z

1 minus βn( 1113857 middotρnfn wn( 1113857

nablafn wn( 1113857

+ βnu minus z⟶ 0

(91)

Since (xnj+1 minus βnju) isin Cnj

we have

c wnj1113874 1113875 +langξnj

xnj+1 minus βnju minus wnjrang +

α2

xnj+1 minus βnju minus wnj

2le 0

(92)

where ξnjisinzc(wnj

) and by the boundedness of zc we get

c wnj1113874 1113875le ξnj

middot wnjminus xnj+1 + βnj

u

minusα2

xnj+1 minus wnjminus βnj

u

2

le ξnj

middot wnjminus xnj+1

+ βnju1113876 1113877

minusα2

xnj+1 minus wnj

2

+ β2nju

2minus 2langxnj+1 minus wnj

βnjurang1113876 1113877⟶ 0

(93)

and using the weak lower semicontinuity of c

c xlowast

( 1113857le lim infj⟶infin

c wnj1113874 1113875le 0 (94)

)us xlowast isin C then xlowast isin S that is ωw(xn) sub SNext we have

xn+1 minus xn

le 1 minus βn( 1113857 wn minus τnnablafn wn( 1113857 minus xn

+ βn u minus xn

le 1 minus βn( 1113857 wn minus xn

+ τn nablafn wn( 1113857

1113960 1113961 + βn xn minus u

le wn minus xn

+

ρnfn wn( 1113857

nablafn wn( 1113857

+ βn xn minus u

⟶ 0

(95)

For z PSu and xnkxlowast isin S using Lemma 1

langu minus z xlowast minus zrangle 0 so

lim supn⟶infinlangu minus z xn minus zrang lim sup

k⟶infinlangu minus z xnk

minus zrang

langu minus z xlowast

minus zrang le 0

(96)

and then

lim supn⟶infinlangu minus z xn+1 minus zrang

lim supn⟶infinlangu minus z xn+1 minus xnrang +langu minus z xn minus zrang( 1113857le 0

(97)

and thus

lim supk

δnk lim sup

k

θnk

βnk

xnkminus xnkminus 1

2

+ 2M11113890

middotθnk

βnk

xnkminus xnkminus 1

+ 2langu minus z xnk+1 minus zrang1113891le 0

(98)

From (82) (83) (98) and Lemma 8 we conclude that thesequence xn1113864 1113865 converges strongly to z PSu )e proof iscomplete

4 Numerical Experiments

In this section we present a numerical experiment to il-lustrate the performance of the proposed algorithms Ournumerical experiments are coded in MATLAB R2007running on personal computer with 350GHz Intel Core i3and 4GB RAM In what follows we apply our algorithms tosolve the problem of least absolute shrinkage and selectionoperator which requires solving a convex optimizationproblem as

minxisinRn

12Ax minus y

2

st x1 le t0

(99)

where A isin Rmtimesn y isin Rm and t0 gt 0 are given elements Inour experiment we first generate an m times n matrix A ran-domly by a standardized normal distribution and x is asparse signal with n elements only K of which is nonzerowhich is also generated randomly )e observation y isgenerated as y Ax )e parameters in this experiment areset with n 512 m 256 ε 10minus 4 and t0 K In thissituation it is readily seen that C x isin Rn c(x)le 0 withc(x) x1 minus t0 and Q y1113864 1113865 which in turn implies that

Cn x isin Rn langξn xrang le langξn wnrang minus wn

1 + t01113966 1113967 (100)

where ξn is defined by

ξn( 1113857i

1 if ξn( 1113857igt 0

[minus 1 1] if ξn( 1113857i 0

minus 1 if ξn( 1113857ilt 0

⎧⎪⎪⎨

⎪⎪⎩(101)

standing for the subdifferential of middot 1 As a half-space theassociated projection onto Cn takes the following form

10 Journal of Mathematics

PCn(x)

x +langξn wn minus xrang minus wn

1 + t0

ξn

2 ξn if x notin Cn

x if x isin Cn

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(102)

To show the efficiency of our algorithm we compareit with the algorithm proposed in [40] )e only dif-ference of these two algorithms is that there are no in-ertial terms in the algorithm proposed in [40] For theconvenience we denote Algorithm 1 by Algo I and thealgorithm in [40] by Algo II respectively In Algo-rithm 1 we set

θn

min 081

n2

xn minus xnminus 1

2

⎧⎨

⎩

⎫⎬

⎭ if xn ne xnminus 1

08 if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

τn

1n Alowast

Awn minus y( 1113857

if A

lowastAwn minus y( 1113857

ne 0

0 if Alowast

Awn minus y( 1113857

0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(103)

In Algo II we set θn equiv 0 and τn is chosen the same asabove )e stopping criterion is that xk+1 minus xklt ε )einitial points are x0 (0 0 0)T andx1 100(1 1 1)T )e numerical results of these twoalgorithms with different choices of the sparsity number K

are listed in Figures 1ndash4 It is easy to see that Algo Iconverges faster than Algo II does which indicates that ourmodified algorithm indeed accelerates the convergence ofthe original algorithm

106

104

102

100

||xR

ndash x R

+1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 1 Iterative results with K 50

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 2 Iterative results with K 40

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 3 Iterative results with K 30

104

102

100

||xR

ndash x R

ndash1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 4 Iterative results with K 20

Journal of Mathematics 11

5 Conclusions

In this paper we present two inertial relaxed CQ algorithmsfor solving split feasibility problems in Hilbert spaces byadopting variable step size )ese algorithms adopt the newconvex subset form and it is easy to calculate the projectionsonto these sets Furthermore step size selection in the al-gorithms does not depend on the operator norm )econvergence theorems are established under some mildconditions and a numerical experiment is given to illustratethe performance of the proposed algorithms

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was partially supported by the National NaturalScience Foundation of China (no 11771126)

References

[1] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquo Numerical Algo-rithms vol 8 no 2 pp 221ndash239 1994

[2] C Byrne ldquoA unified treatment of some iterative algorithms insignal processing and image reconstructionrdquo Inverse Prob-lems vol 20 no 1 pp 103ndash120 2004

[3] Y Censor T Elfving N Kopf and T Bortfeld ldquo)emultiple-sets split feasibility problem and its applications for inverseproblemsrdquo Inverse Problems vol 21 no 6 pp 2071ndash20842005

[4] C Byrne ldquoIterative oblique projection onto convex sets andthe split feasibility problemrdquo Inverse Problems vol 18 no 2pp 441ndash453 2002

[5] S He H Tian and H K Xu ldquo)e selective projection methodfor convex feasibility and split feasibility problemsrdquo Journal ofNonlinear and Convex Analysis vol 19 pp 1199ndash1215 2018

[6] S He Z Zhao and B Luo ldquoA relaxed self-adaptive CQ al-gorithm for the multiple-sets split feasibility problemrdquo Op-timization vol 64 no 9 pp 1907ndash1918 2015

[7] G L Acedo V Martin-Marquez F Wang and H XuldquoSolving the split feasibility problem without prior knowledgeof matrix normsrdquo Inverse Problems vol 28 Article ID 0850042012

[8] B Qu and N Xiu ldquoA new halfspace-relaxation projectionmethod for the split feasibility problemrdquo Linear Algebra andits Applications vol 428 no 5-6 pp 1218ndash1229 2008

[9] F Wang ldquoA splitting-relaxed projection method for solvingthe split feasibility problemrdquo Fixed Point gteory vol 14pp 211ndash218 2013

[10] F Wang ldquoOn the convergence of CQ algorithm with variablesteps for the split equality problemrdquo Numerical Algorithmsvol 74 no 3 pp 927ndash935 2017

[11] F Wang ldquoPolyakrsquos gradient method for split feasibilityproblem constrained by level setsrdquo Numerical Algorithmsvol 77 no 3 pp 925ndash938 2018

[12] H-K Xu ldquoA variable Krasnoselrsquoskii-Mann algorithm and themultiple-set split feasibility problemrdquo Inverse Problemsvol 22 no 6 pp 2021ndash2034 2006

[13] Q Yang ldquo)e relaxed CQ algorithm solving the split feasi-bility problemrdquo Inverse Problems vol 20 no 4pp 1261ndash1266 2004

[14] F Alvarez and H Attouch ldquoAn inertial proximal method formaximal monotone operators via discretization of a nonlinearoscillator with dampingrdquo Set-Valued Analysis vol 9 pp 3ndash112001

[15] Y Dang J Sun J Sun and H Xu ldquoInertial accelerated al-gorithms for solving a split feasibility problemrdquo Journal ofIndustrial amp Management Optimization vol 13 no 3pp 1383ndash1394 2017

[16] A Gibali D Mai D )i Mai and N )e Vinh ldquoA newrelaxed CQ algorithm for solving split feasibility problems inHilbert spaces and its applicationsrdquo Journal of Industrial ampManagement Optimization vol 15 no 2 pp 963ndash984 2019

[17] Q L Dong Y J Cho and T M Rassias ldquoGeneral inertialMann algorithms and their convergence analysis for non-expansive mappingsrdquo in Applications of Nonlinear Analysispp 175ndash191 Springer Berlin Germany 2018

[18] Q L Dong Y J Cho L L Zhong and T M Rassias ldquoInertialprojection and contraction algorithms for variational in-equalitiesrdquo Journal of Global Optimization vol 70 no 3pp 687ndash704 2018

[19] Q L Dong J Z Huang X H Li Y J Cho and T M RassiasldquoMiKM multi-step inertial Krasnoselrsquoskiǐ-Mann algorithmand its applicationsrdquo Journal of Global Optimization vol 73no 4 pp 801ndash824 2019

[20] Q L Dong H B Yuan Y J Cho and T M RassiasldquoModified inertial Mann algorithm and inertial CQ-algorithmfor nonexpansive mappingsrdquo Optimization Letters vol 12no 1 pp 87ndash102 2018

[21] Y Shehu and A Gibali ldquoNew inertial relaxed method forsolving split feasibilitiesrdquo Optimization Letters 2020

[22] S S Tai N Pholasa and P Cholamjiak ldquo)emodified inertialrelaxed CQ algorithm for solving the split feasibility prob-lemsrdquo Journal of Industrial and Management Optimizationvol 14 pp 1595ndash1615 2018

[23] S S Tai N Pholasa and P Cholamjiak ldquoRelaxed CQ algo-rithms involving the inertial technique for multiple-sets splitfeasibility problemsrdquo Revista de la Real Academia de CienciasExactas Fisicas y Naturales-Serie A Matematicas vol 113pp 1081ndash1099 2019

[24] S Y Cho ldquoA convergence theorem for generalized mixedequilibrium problems and multivalued asymptotically non-expansive mappingsrdquo Journal of Nonlinear and ConvexAnalysis vol 21 pp 1017ndash1026 2020

[25] H-K Xu ldquoIterative methods for the split feasibility problemin infinite-dimensional Hilbert spacesrdquo Inverse Problemsvol 26 no 10 Article ID 105018 2010

[26] F Wang and H Xu ldquoApproximating curve and strongconvergence of the CQ algorithm for the split feasibilityproblemrdquo Journal of Inequalities and Applications vol 2010Article ID 102085 2010

[27] M Tian and H F Zhang ldquo)e regularized CQ algorithmwithout a priori knowledge of operator norm for solving thesplit feasibility problemrdquo Journal of Inequalities and Appli-cations vol 2017 p 207 2017

[28] Y H Yao W J Gang and Y C Liou ldquoRegularized methodsfor the split feasibility problemrdquo Abstract and AppliedAnalysis vol 2012 Article ID 140679 2012

12 Journal of Mathematics

[29] Y Zhou Z Haiyun and P Wang ldquoIterative methods forfinding the minimum-norm solution of the standardmonotone variational inequality problems with applicationsin Hilbert spacesrdquo Journal of Inequalities and Applicationsvol 2015 2015

[30] T V Nguyen C Prasit and S Suthep ldquoA new CQ algorithmfor solving split feasibility problems in Hilbert spacesrdquoMalaysian Mathematical Sciences Society and Penerbit Uni-versiti Sains Malaysia vol 42 no 5 2020

[31] D R Sahu Y J Cho Q L Dong M R Kashyap and X H LiInertial Relaxed CQ algorithms for Solving a Split FeasibilityProblem in Hilbert Spaces Springer Science Business MediaBerlin Germany 2020

[32] R P Agarwal D O Regan and D R Sahu ldquoFixed pointtheory for Lipschitzian-type mappings with applicationsrdquo inTopological Fixed Point gteory and Its Applications SpringerNew York NY USA 2009

[33] S Y Cho ldquoAmonotone Bregan projection algorithm for fixedpoint and equilibrium problems in a relative Banach spacerdquoFilomat vol 34 pp 1487ndash1497 2020

[34] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator gteory in Hilbert Spaces Springer NewYork NY USA 2011

[35] P E Mainge ldquoInertial iterative process for fixed points ofcertain quasi-nonexpansive mappingrdquo Set Valued Analysisvol 15 pp 67ndash79 2007

[36] P E Mainge ldquoApproximation methods for common fixedpoints of nonexpansive mappings in Hilbert spacesrdquo Journalof Mathematical Analysis and Applications vol 325pp 469ndash479 2007

[37] A Moudafi and A Gibali ldquoregularization of split feasibilityproblemsrdquo Numerical Algorithmsvol 78 no 3 pp 1ndash192017

[38] S He and C Yang ldquoSolving the variational inequality problemdefined on intersection of finite level setsrdquo Abstract andApplied Analysis vol 2013 Article ID 942315 2013

[39] F Wang and H Yu ldquoAn inertial relaxed CQ algorithm withan application to the LASSO and elastic netrdquo Optimizationvol 2020 Article ID 1763989 2020

[40] P K Anh N T Vinh and V T Dung ldquoA new self-adaptiveCQ algorithm with an application to the LASSO problemrdquoJournal of Fixed Point gteory and Applications vol 20 p 1422018

Journal of Mathematics 13

PCn(x)

x +langξn wn minus xrang minus wn

1 + t0

ξn

2 ξn if x notin Cn

x if x isin Cn

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(102)

To show the efficiency of our algorithm we compareit with the algorithm proposed in [40] )e only dif-ference of these two algorithms is that there are no in-ertial terms in the algorithm proposed in [40] For theconvenience we denote Algorithm 1 by Algo I and thealgorithm in [40] by Algo II respectively In Algo-rithm 1 we set

θn

min 081

n2

xn minus xnminus 1

2

⎧⎨

⎩

⎫⎬

⎭ if xn ne xnminus 1

08 if xn xnminus 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

τn

1n Alowast

Awn minus y( 1113857

if A

lowastAwn minus y( 1113857

ne 0

0 if Alowast

Awn minus y( 1113857

0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(103)

In Algo II we set θn equiv 0 and τn is chosen the same asabove )e stopping criterion is that xk+1 minus xklt ε )einitial points are x0 (0 0 0)T andx1 100(1 1 1)T )e numerical results of these twoalgorithms with different choices of the sparsity number K

are listed in Figures 1ndash4 It is easy to see that Algo Iconverges faster than Algo II does which indicates that ourmodified algorithm indeed accelerates the convergence ofthe original algorithm

106

104

102

100

||xR

ndash x R

+1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 1 Iterative results with K 50

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 2 Iterative results with K 40

104

102

103

101

100

10ndash1||xR

ndash x R

+1||

10ndash2

10ndash4

10ndash3

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 3 Iterative results with K 30

104

102

100

||xR

ndash x R

ndash1||

10ndash2

10ndash4

0 100 200 300 400Iteration numbers

500 600 700 800 900 1000

ALGO IALGO II

Figure 4 Iterative results with K 20

Journal of Mathematics 11

5 Conclusions

In this paper we present two inertial relaxed CQ algorithmsfor solving split feasibility problems in Hilbert spaces byadopting variable step size )ese algorithms adopt the newconvex subset form and it is easy to calculate the projectionsonto these sets Furthermore step size selection in the al-gorithms does not depend on the operator norm )econvergence theorems are established under some mildconditions and a numerical experiment is given to illustratethe performance of the proposed algorithms

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was partially supported by the National NaturalScience Foundation of China (no 11771126)

References

[1] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquo Numerical Algo-rithms vol 8 no 2 pp 221ndash239 1994

[2] C Byrne ldquoA unified treatment of some iterative algorithms insignal processing and image reconstructionrdquo Inverse Prob-lems vol 20 no 1 pp 103ndash120 2004

[3] Y Censor T Elfving N Kopf and T Bortfeld ldquo)emultiple-sets split feasibility problem and its applications for inverseproblemsrdquo Inverse Problems vol 21 no 6 pp 2071ndash20842005

[4] C Byrne ldquoIterative oblique projection onto convex sets andthe split feasibility problemrdquo Inverse Problems vol 18 no 2pp 441ndash453 2002

[5] S He H Tian and H K Xu ldquo)e selective projection methodfor convex feasibility and split feasibility problemsrdquo Journal ofNonlinear and Convex Analysis vol 19 pp 1199ndash1215 2018

[6] S He Z Zhao and B Luo ldquoA relaxed self-adaptive CQ al-gorithm for the multiple-sets split feasibility problemrdquo Op-timization vol 64 no 9 pp 1907ndash1918 2015

[7] G L Acedo V Martin-Marquez F Wang and H XuldquoSolving the split feasibility problem without prior knowledgeof matrix normsrdquo Inverse Problems vol 28 Article ID 0850042012

[8] B Qu and N Xiu ldquoA new halfspace-relaxation projectionmethod for the split feasibility problemrdquo Linear Algebra andits Applications vol 428 no 5-6 pp 1218ndash1229 2008

[9] F Wang ldquoA splitting-relaxed projection method for solvingthe split feasibility problemrdquo Fixed Point gteory vol 14pp 211ndash218 2013

[10] F Wang ldquoOn the convergence of CQ algorithm with variablesteps for the split equality problemrdquo Numerical Algorithmsvol 74 no 3 pp 927ndash935 2017

[11] F Wang ldquoPolyakrsquos gradient method for split feasibilityproblem constrained by level setsrdquo Numerical Algorithmsvol 77 no 3 pp 925ndash938 2018

[12] H-K Xu ldquoA variable Krasnoselrsquoskii-Mann algorithm and themultiple-set split feasibility problemrdquo Inverse Problemsvol 22 no 6 pp 2021ndash2034 2006

[13] Q Yang ldquo)e relaxed CQ algorithm solving the split feasi-bility problemrdquo Inverse Problems vol 20 no 4pp 1261ndash1266 2004

[14] F Alvarez and H Attouch ldquoAn inertial proximal method formaximal monotone operators via discretization of a nonlinearoscillator with dampingrdquo Set-Valued Analysis vol 9 pp 3ndash112001

[15] Y Dang J Sun J Sun and H Xu ldquoInertial accelerated al-gorithms for solving a split feasibility problemrdquo Journal ofIndustrial amp Management Optimization vol 13 no 3pp 1383ndash1394 2017

[16] A Gibali D Mai D )i Mai and N )e Vinh ldquoA newrelaxed CQ algorithm for solving split feasibility problems inHilbert spaces and its applicationsrdquo Journal of Industrial ampManagement Optimization vol 15 no 2 pp 963ndash984 2019

[17] Q L Dong Y J Cho and T M Rassias ldquoGeneral inertialMann algorithms and their convergence analysis for non-expansive mappingsrdquo in Applications of Nonlinear Analysispp 175ndash191 Springer Berlin Germany 2018

[18] Q L Dong Y J Cho L L Zhong and T M Rassias ldquoInertialprojection and contraction algorithms for variational in-equalitiesrdquo Journal of Global Optimization vol 70 no 3pp 687ndash704 2018

[19] Q L Dong J Z Huang X H Li Y J Cho and T M RassiasldquoMiKM multi-step inertial Krasnoselrsquoskiǐ-Mann algorithmand its applicationsrdquo Journal of Global Optimization vol 73no 4 pp 801ndash824 2019

[20] Q L Dong H B Yuan Y J Cho and T M RassiasldquoModified inertial Mann algorithm and inertial CQ-algorithmfor nonexpansive mappingsrdquo Optimization Letters vol 12no 1 pp 87ndash102 2018

[21] Y Shehu and A Gibali ldquoNew inertial relaxed method forsolving split feasibilitiesrdquo Optimization Letters 2020

[22] S S Tai N Pholasa and P Cholamjiak ldquo)emodified inertialrelaxed CQ algorithm for solving the split feasibility prob-lemsrdquo Journal of Industrial and Management Optimizationvol 14 pp 1595ndash1615 2018

[23] S S Tai N Pholasa and P Cholamjiak ldquoRelaxed CQ algo-rithms involving the inertial technique for multiple-sets splitfeasibility problemsrdquo Revista de la Real Academia de CienciasExactas Fisicas y Naturales-Serie A Matematicas vol 113pp 1081ndash1099 2019

[24] S Y Cho ldquoA convergence theorem for generalized mixedequilibrium problems and multivalued asymptotically non-expansive mappingsrdquo Journal of Nonlinear and ConvexAnalysis vol 21 pp 1017ndash1026 2020

[25] H-K Xu ldquoIterative methods for the split feasibility problemin infinite-dimensional Hilbert spacesrdquo Inverse Problemsvol 26 no 10 Article ID 105018 2010

[26] F Wang and H Xu ldquoApproximating curve and strongconvergence of the CQ algorithm for the split feasibilityproblemrdquo Journal of Inequalities and Applications vol 2010Article ID 102085 2010

[27] M Tian and H F Zhang ldquo)e regularized CQ algorithmwithout a priori knowledge of operator norm for solving thesplit feasibility problemrdquo Journal of Inequalities and Appli-cations vol 2017 p 207 2017

[28] Y H Yao W J Gang and Y C Liou ldquoRegularized methodsfor the split feasibility problemrdquo Abstract and AppliedAnalysis vol 2012 Article ID 140679 2012

12 Journal of Mathematics

[29] Y Zhou Z Haiyun and P Wang ldquoIterative methods forfinding the minimum-norm solution of the standardmonotone variational inequality problems with applicationsin Hilbert spacesrdquo Journal of Inequalities and Applicationsvol 2015 2015

[30] T V Nguyen C Prasit and S Suthep ldquoA new CQ algorithmfor solving split feasibility problems in Hilbert spacesrdquoMalaysian Mathematical Sciences Society and Penerbit Uni-versiti Sains Malaysia vol 42 no 5 2020

[31] D R Sahu Y J Cho Q L Dong M R Kashyap and X H LiInertial Relaxed CQ algorithms for Solving a Split FeasibilityProblem in Hilbert Spaces Springer Science Business MediaBerlin Germany 2020

[32] R P Agarwal D O Regan and D R Sahu ldquoFixed pointtheory for Lipschitzian-type mappings with applicationsrdquo inTopological Fixed Point gteory and Its Applications SpringerNew York NY USA 2009

[33] S Y Cho ldquoAmonotone Bregan projection algorithm for fixedpoint and equilibrium problems in a relative Banach spacerdquoFilomat vol 34 pp 1487ndash1497 2020

[34] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator gteory in Hilbert Spaces Springer NewYork NY USA 2011

[35] P E Mainge ldquoInertial iterative process for fixed points ofcertain quasi-nonexpansive mappingrdquo Set Valued Analysisvol 15 pp 67ndash79 2007

[36] P E Mainge ldquoApproximation methods for common fixedpoints of nonexpansive mappings in Hilbert spacesrdquo Journalof Mathematical Analysis and Applications vol 325pp 469ndash479 2007

[37] A Moudafi and A Gibali ldquoregularization of split feasibilityproblemsrdquo Numerical Algorithmsvol 78 no 3 pp 1ndash192017

[38] S He and C Yang ldquoSolving the variational inequality problemdefined on intersection of finite level setsrdquo Abstract andApplied Analysis vol 2013 Article ID 942315 2013

[39] F Wang and H Yu ldquoAn inertial relaxed CQ algorithm withan application to the LASSO and elastic netrdquo Optimizationvol 2020 Article ID 1763989 2020

[40] P K Anh N T Vinh and V T Dung ldquoA new self-adaptiveCQ algorithm with an application to the LASSO problemrdquoJournal of Fixed Point gteory and Applications vol 20 p 1422018

Journal of Mathematics 13

5 Conclusions

In this paper we present two inertial relaxed CQ algorithmsfor solving split feasibility problems in Hilbert spaces byadopting variable step size )ese algorithms adopt the newconvex subset form and it is easy to calculate the projectionsonto these sets Furthermore step size selection in the al-gorithms does not depend on the operator norm )econvergence theorems are established under some mildconditions and a numerical experiment is given to illustratethe performance of the proposed algorithms

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was partially supported by the National NaturalScience Foundation of China (no 11771126)

References

[1] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquo Numerical Algo-rithms vol 8 no 2 pp 221ndash239 1994

[2] C Byrne ldquoA unified treatment of some iterative algorithms insignal processing and image reconstructionrdquo Inverse Prob-lems vol 20 no 1 pp 103ndash120 2004

[3] Y Censor T Elfving N Kopf and T Bortfeld ldquo)emultiple-sets split feasibility problem and its applications for inverseproblemsrdquo Inverse Problems vol 21 no 6 pp 2071ndash20842005

[4] C Byrne ldquoIterative oblique projection onto convex sets andthe split feasibility problemrdquo Inverse Problems vol 18 no 2pp 441ndash453 2002

[5] S He H Tian and H K Xu ldquo)e selective projection methodfor convex feasibility and split feasibility problemsrdquo Journal ofNonlinear and Convex Analysis vol 19 pp 1199ndash1215 2018

[6] S He Z Zhao and B Luo ldquoA relaxed self-adaptive CQ al-gorithm for the multiple-sets split feasibility problemrdquo Op-timization vol 64 no 9 pp 1907ndash1918 2015

[7] G L Acedo V Martin-Marquez F Wang and H XuldquoSolving the split feasibility problem without prior knowledgeof matrix normsrdquo Inverse Problems vol 28 Article ID 0850042012

[8] B Qu and N Xiu ldquoA new halfspace-relaxation projectionmethod for the split feasibility problemrdquo Linear Algebra andits Applications vol 428 no 5-6 pp 1218ndash1229 2008

[9] F Wang ldquoA splitting-relaxed projection method for solvingthe split feasibility problemrdquo Fixed Point gteory vol 14pp 211ndash218 2013

[10] F Wang ldquoOn the convergence of CQ algorithm with variablesteps for the split equality problemrdquo Numerical Algorithmsvol 74 no 3 pp 927ndash935 2017

[11] F Wang ldquoPolyakrsquos gradient method for split feasibilityproblem constrained by level setsrdquo Numerical Algorithmsvol 77 no 3 pp 925ndash938 2018

[12] H-K Xu ldquoA variable Krasnoselrsquoskii-Mann algorithm and themultiple-set split feasibility problemrdquo Inverse Problemsvol 22 no 6 pp 2021ndash2034 2006

[13] Q Yang ldquo)e relaxed CQ algorithm solving the split feasi-bility problemrdquo Inverse Problems vol 20 no 4pp 1261ndash1266 2004

[14] F Alvarez and H Attouch ldquoAn inertial proximal method formaximal monotone operators via discretization of a nonlinearoscillator with dampingrdquo Set-Valued Analysis vol 9 pp 3ndash112001

[15] Y Dang J Sun J Sun and H Xu ldquoInertial accelerated al-gorithms for solving a split feasibility problemrdquo Journal ofIndustrial amp Management Optimization vol 13 no 3pp 1383ndash1394 2017

[16] A Gibali D Mai D )i Mai and N )e Vinh ldquoA newrelaxed CQ algorithm for solving split feasibility problems inHilbert spaces and its applicationsrdquo Journal of Industrial ampManagement Optimization vol 15 no 2 pp 963ndash984 2019

[17] Q L Dong Y J Cho and T M Rassias ldquoGeneral inertialMann algorithms and their convergence analysis for non-expansive mappingsrdquo in Applications of Nonlinear Analysispp 175ndash191 Springer Berlin Germany 2018

[18] Q L Dong Y J Cho L L Zhong and T M Rassias ldquoInertialprojection and contraction algorithms for variational in-equalitiesrdquo Journal of Global Optimization vol 70 no 3pp 687ndash704 2018

[19] Q L Dong J Z Huang X H Li Y J Cho and T M RassiasldquoMiKM multi-step inertial Krasnoselrsquoskiǐ-Mann algorithmand its applicationsrdquo Journal of Global Optimization vol 73no 4 pp 801ndash824 2019

[20] Q L Dong H B Yuan Y J Cho and T M RassiasldquoModified inertial Mann algorithm and inertial CQ-algorithmfor nonexpansive mappingsrdquo Optimization Letters vol 12no 1 pp 87ndash102 2018

[21] Y Shehu and A Gibali ldquoNew inertial relaxed method forsolving split feasibilitiesrdquo Optimization Letters 2020

[22] S S Tai N Pholasa and P Cholamjiak ldquo)emodified inertialrelaxed CQ algorithm for solving the split feasibility prob-lemsrdquo Journal of Industrial and Management Optimizationvol 14 pp 1595ndash1615 2018

[23] S S Tai N Pholasa and P Cholamjiak ldquoRelaxed CQ algo-rithms involving the inertial technique for multiple-sets splitfeasibility problemsrdquo Revista de la Real Academia de CienciasExactas Fisicas y Naturales-Serie A Matematicas vol 113pp 1081ndash1099 2019

[24] S Y Cho ldquoA convergence theorem for generalized mixedequilibrium problems and multivalued asymptotically non-expansive mappingsrdquo Journal of Nonlinear and ConvexAnalysis vol 21 pp 1017ndash1026 2020

[25] H-K Xu ldquoIterative methods for the split feasibility problemin infinite-dimensional Hilbert spacesrdquo Inverse Problemsvol 26 no 10 Article ID 105018 2010

[26] F Wang and H Xu ldquoApproximating curve and strongconvergence of the CQ algorithm for the split feasibilityproblemrdquo Journal of Inequalities and Applications vol 2010Article ID 102085 2010

[27] M Tian and H F Zhang ldquo)e regularized CQ algorithmwithout a priori knowledge of operator norm for solving thesplit feasibility problemrdquo Journal of Inequalities and Appli-cations vol 2017 p 207 2017

[28] Y H Yao W J Gang and Y C Liou ldquoRegularized methodsfor the split feasibility problemrdquo Abstract and AppliedAnalysis vol 2012 Article ID 140679 2012

12 Journal of Mathematics

[29] Y Zhou Z Haiyun and P Wang ldquoIterative methods forfinding the minimum-norm solution of the standardmonotone variational inequality problems with applicationsin Hilbert spacesrdquo Journal of Inequalities and Applicationsvol 2015 2015

[30] T V Nguyen C Prasit and S Suthep ldquoA new CQ algorithmfor solving split feasibility problems in Hilbert spacesrdquoMalaysian Mathematical Sciences Society and Penerbit Uni-versiti Sains Malaysia vol 42 no 5 2020

[31] D R Sahu Y J Cho Q L Dong M R Kashyap and X H LiInertial Relaxed CQ algorithms for Solving a Split FeasibilityProblem in Hilbert Spaces Springer Science Business MediaBerlin Germany 2020

[32] R P Agarwal D O Regan and D R Sahu ldquoFixed pointtheory for Lipschitzian-type mappings with applicationsrdquo inTopological Fixed Point gteory and Its Applications SpringerNew York NY USA 2009

[33] S Y Cho ldquoAmonotone Bregan projection algorithm for fixedpoint and equilibrium problems in a relative Banach spacerdquoFilomat vol 34 pp 1487ndash1497 2020

[34] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator gteory in Hilbert Spaces Springer NewYork NY USA 2011

[35] P E Mainge ldquoInertial iterative process for fixed points ofcertain quasi-nonexpansive mappingrdquo Set Valued Analysisvol 15 pp 67ndash79 2007

[36] P E Mainge ldquoApproximation methods for common fixedpoints of nonexpansive mappings in Hilbert spacesrdquo Journalof Mathematical Analysis and Applications vol 325pp 469ndash479 2007

[37] A Moudafi and A Gibali ldquoregularization of split feasibilityproblemsrdquo Numerical Algorithmsvol 78 no 3 pp 1ndash192017

[38] S He and C Yang ldquoSolving the variational inequality problemdefined on intersection of finite level setsrdquo Abstract andApplied Analysis vol 2013 Article ID 942315 2013

[39] F Wang and H Yu ldquoAn inertial relaxed CQ algorithm withan application to the LASSO and elastic netrdquo Optimizationvol 2020 Article ID 1763989 2020

[40] P K Anh N T Vinh and V T Dung ldquoA new self-adaptiveCQ algorithm with an application to the LASSO problemrdquoJournal of Fixed Point gteory and Applications vol 20 p 1422018

Journal of Mathematics 13

[29] Y Zhou Z Haiyun and P Wang ldquoIterative methods forfinding the minimum-norm solution of the standardmonotone variational inequality problems with applicationsin Hilbert spacesrdquo Journal of Inequalities and Applicationsvol 2015 2015

[30] T V Nguyen C Prasit and S Suthep ldquoA new CQ algorithmfor solving split feasibility problems in Hilbert spacesrdquoMalaysian Mathematical Sciences Society and Penerbit Uni-versiti Sains Malaysia vol 42 no 5 2020

[31] D R Sahu Y J Cho Q L Dong M R Kashyap and X H LiInertial Relaxed CQ algorithms for Solving a Split FeasibilityProblem in Hilbert Spaces Springer Science Business MediaBerlin Germany 2020

[32] R P Agarwal D O Regan and D R Sahu ldquoFixed pointtheory for Lipschitzian-type mappings with applicationsrdquo inTopological Fixed Point gteory and Its Applications SpringerNew York NY USA 2009

[33] S Y Cho ldquoAmonotone Bregan projection algorithm for fixedpoint and equilibrium problems in a relative Banach spacerdquoFilomat vol 34 pp 1487ndash1497 2020

[34] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator gteory in Hilbert Spaces Springer NewYork NY USA 2011

[35] P E Mainge ldquoInertial iterative process for fixed points ofcertain quasi-nonexpansive mappingrdquo Set Valued Analysisvol 15 pp 67ndash79 2007

[36] P E Mainge ldquoApproximation methods for common fixedpoints of nonexpansive mappings in Hilbert spacesrdquo Journalof Mathematical Analysis and Applications vol 325pp 469ndash479 2007

[37] A Moudafi and A Gibali ldquoregularization of split feasibilityproblemsrdquo Numerical Algorithmsvol 78 no 3 pp 1ndash192017

[38] S He and C Yang ldquoSolving the variational inequality problemdefined on intersection of finite level setsrdquo Abstract andApplied Analysis vol 2013 Article ID 942315 2013

[39] F Wang and H Yu ldquoAn inertial relaxed CQ algorithm withan application to the LASSO and elastic netrdquo Optimizationvol 2020 Article ID 1763989 2020

[40] P K Anh N T Vinh and V T Dung ldquoA new self-adaptiveCQ algorithm with an application to the LASSO problemrdquoJournal of Fixed Point gteory and Applications vol 20 p 1422018

Journal of Mathematics 13