a class of hybrid methods for quasi-variational inequalities

Optim LettDOI 10.1007/s11590-014-0729-7

ORIGINAL PAPER

A class of hybrid methods for quasi-variationalinequalities

Thi Thu Van Nguyen · Thi Phuong Dong Nguyen ·Jean Jacques Strodiot · Van Hien Nguyen

Received: 3 October 2013 / Accepted: 29 January 2014© Springer-Verlag Berlin Heidelberg 2014

Abstract In this paper we develop a new and efficient method for solving a quasi-variational inequality problem (QVIP) by using an extragradient-type method. Thestrategy is to combine the well-known search directions in the correction step fromliterature with the direction defined by the current iterate and the trial point obtainedin the prediction step. This new combined search direction allows us to improve theconvergence of the sequence of iterates to the solution of the QVIP but under a slightlystronger assumption, namely the co-coercivity of the problem operator. The new algo-rithm is devised to solve problems where the projections onto the moving feasible setare not easy to obtain. This combined procedure is applied to three well-known searchdirections and numerical illustrations are given to show the improvements obtainedthanks to this strategy.

Keywords Quasi-variational inequalities · Hybrid extragradient methods ·Generalized Nash equilibrium problems · Two-step methods · Co-coercivity

T. T. V. NguyenInstitute for Computational Science and Technology (ICST) and University of Science,VNU-HCM, Ho Chi Minh City, Vietname-mail: [email protected]

T. P. D. Nguyen · J. J. Strodiot· V. H. NguyenInstitute for Computational Science and Technology (ICST), Ho Chi Minh City, Vietnam

J. J. Strodiot (B) · V. H. NguyenUniversity of Namur, Namur, Belgiume-mail: [email protected]

T. P. D. Nguyene-mail: [email protected]

V. H. Nguyene-mail: [email protected]

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T. T. Van Nguyen et al.

1 Introduction

Quasi-variational inequality problems, QVIPs for short, have recently received agreat deal of attention in the research community. Many important problems canbe reformulated as QVIPs. The connection between the generalized Nash games andquasi-variational inequalities was recognized by Bensoussan [1] as early as 1974.Harker [8] revisited these problems in Euclidean spaces. Kocvara and Outrata [9]discussed a class of quasi-variational inequalities with applications to engineering.Wei and Smeers [20] introduced a quasi-variational inequality formulation of a spatialoligopolistic electricity model with Cournot generators and regulated transmissionprices. Recently, Scrimali [17] shows how the so-called joint implementation in envi-ronmental projects can be studied as an infinite-dimensional quasi-variational inequal-ity, and Lenzen et al. [10] introduce a class of quasi-variational inequalities for adaptiveimage denoising and decomposition. The QVIP is often studied in connection withthe Generalized Nash equilibrium problem, the equilibrium problem and the quasi-equilibrium problem; for more details, see for instance [7,8,15,16,18,21]. Our aim inthis paper is to introduce and study a new class of projection-like methods for solvingQVIP.

Recently, Zhang et al. [21] proposed a two-step projection method for QVIPs con-sidering at each iteration first a prediction step and then a correction step. They devel-oped two procedures for computing the prediction step depending on the difficultyto calculate orthogonal projections onto the moving feasible set. When the projec-tion is easy to obtain, the steplength associated with the direction is reduced until animprovement criterion is satisfied. This procedure supposes that at each updating ofthe steplength a new projection is done. When the projection onto the moving feasibleset is numerically expensive, it is preferable to calculate first a single projection ontothe feasible set to get a trial point, and afterwards, to perform a linesearch procedurebetween the current point and the trial point to obtain the prediction step. Once thisstep has been calculated, the correction step is obtained thanks to a search direction anda steplength. Very recently, Han et al. [7] reconsidered the case when the projectiononto the feasible set is easy to calculate. In fact, they have replaced the search direc-tion used by Zhang et al. by a convex combination of this direction with the directiondefined by the trial point obtained earlier in the prediction step. They have obtaineda new algorithm whose convergence is proven under the assumption that the operatordefining the QVIP is co-coercive. Moreover they have also shown that the numericalbehavior of the new algorithm is much better than the one obtained in [21].

The aim of this paper is to develop an algorithm similar to the one proposed by Hanet al. [7] but when it is not assumed that the projection onto the feasible set is easy tocalculate. More precisely, we replace the prediction step used by Han et al. by anotherprediction step where only one projection is done to get a trial point and where thenext point is obtained by a linesearch between this trial point and the current iterate.Moreover, in the correction step we propose to combine the direction defined by thetrial point and the current point with one of the three well-known directions used inthe literature. The convergence is obtained under a general property satisfied by all ofthese three directions. Let us mention here that such an idea has been used in [6] forsolving a variational inequality problem (VIP).

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Quasi-variational inequalities

The remainder of the paper is organized as follows. In Sect. 2 we state the QVIformulation formally and summarize some definitions, properties and results that willbe used in the sequel. In Sect. 3, we introduce the general method for solving theQVIP and prove its convergence. Some preliminary numerical results are presentedin Sect. 4 while some concluding remarks are given in Sect. 5.

2 Preliminaries

Let X be a nonempty, closed and convex subset of Rn , F be a monotone operator from

X to Rn , and K be a multivalued operator from X into itself verifying for every x ∈ X

the two properties: x ∈ K (x) and K (x) is a nonempty, closed and convex subset ofX . We consider the QVIP which is to find a point x∗ ∈ K (x∗) such that

〈F(x∗), y − x∗〉 ≥ 0 for all y ∈ K (x∗).

When K (x) is a fixed constraint set, say, K (x) = K for all x , the QVIP becomesthe classical VIP of finding x∗ ∈ K such that

〈F(x∗), y − x∗〉 ≥ 0 for all y ∈ K .

There are a lot of computational methods for solving the VIP; see, for example,the monographs of Nagurney [11] and of Facchinei and Pang [2] and the referencestherein. Compared with the VIP, the literature on the algorithms for the QVIP is notas extensive; see, for instance [4,5,7,12,18,21].

Most of the projection methods for solving the QVIP are based on two-step iterations(see, for example, [21]): Each iteration consists in a prediction step followed by acorrection step. More precisely, let xk ∈ X . Two procedures can be used to get thenext iterate xk+1, depending on the numerical difficulty to compute the projection ontothe moving feasible set K (xk). When the projection onto K (xk) is easy to compute,the prediction step can be defined by

xk = PK (xk )(xk − βk F(xk)) (2.1)

where βk = γ lmk , γ ∈ (0, 1), l ∈ (0, 1), and mk is the smallest nonnegative integerm such that

βk〈F(xk) − F(xk), xk − xk〉 ≤ c ||xk − xk ||2

with c ∈ (0, 1). In this procedure, a new projection onto K (xk) must be computedeach time the parameter mk is updated.

When the projection onto K (xk) is numerically more expensive, it is preferable touse only one projection on K (xk) per linesearch. So, in that situation, we first calculate

zk = PK (xk )(xk − F(xk))

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and after, we computeyk = (1 − βk)xk + βk zk (2.2)

where βk = lmk , l ∈ (0, 1), and mk is the smallest nonnegative integer m such that

〈F(xk) − F(yk), xk − zk〉 ≤ c ||xk − zk ||2

where c ∈ (0, 1).Once xk or yk is obtained, a correction step is done by calculating

xk+1 = PK (xk )(xk − αkdk)

where dk is a search direction and αk is a steplength.When xk is used [see (2.1)], Zhang et al. [21] proposed to take [with σ ∈ (0, 2)]

dk = xk − xk + βk F(xk) := d Z1k and αk = σ(1 − c)

||xk − xk ||2||dk ||2

for the search direction and the steplength along this direction, respectively.On the other hand, when it is yk that is used [see (2.2)], these authors suggested to

take [with σ ∈ (0, 2)]

dk = xk − zk + 1

βkF(yk) := d Z

k and αk = σ(1 − c)||xk − zk ||2

||dk ||2 .

With this choice, they proved [[21], Lemma 5.2, inequality (29)] that

〈d Zk , xk − x∗〉 ≥ ‖xk − zk‖2 − 〈F(xk) − F(yk), xk − zk〉. (2.3)

On the other hand, Han et al. [7] recently revisited the prediction step in the casewhen xk is used, and proposed, in the correction step, to combine the direction dk :=d Z1

k − βk F(xk) with the direction xk − xk as follows:

dk = ρdk + (1 − ρ)(xk − xk)

where xk is given by (2.1) and ρ ∈ (0, 1). With this strategy, the numerical behaviorof Han et al.’s algorithm [7] is better than the one of Zhang et al. [21]. However, itsconvergence is obtained under the assumption that F is co-coercive, while Zhang andal.’s algorithm only requires the monotonicity of F to ensure the convergence.

Our aim in this paper is to modify Han and al.’s [7] algorithm as follows: Insteadof computing the prediction step xk given by (2.1), we propose to use the predictionstep yk given by (2.2); so doing, the projection step zk is computed only once (whichwas not the case in [7]). Furthermore, to obtain a very general algorithm, we considera class of search directions which will be used in the correction step. This allows us touse not only the Zhang et al. direction but also two other directions proposed by Nooret al. in [13,14]. The convergence of our general algorithm will be obtained under thefollowing assumption:

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Assumption A (a) S∗ = ∅. Here S∗ is defined as S∗ = {x ∈ S|〈y − x, F(x)〉 ≥0 ∀y ∈ S} where S = ∩x∈X K (x) and S = ∪x∈X K (x)

(b) F is co-coercive with modulus μ on S(c) K is continuous on X .

Let us recall that F is co-coercive with modulus μ > 0 (or μ-co-coercive) on X if,for every x, y ∈ X ,

〈F(y) − F(x), y − x〉 ≥ μ ‖F(y) − F(x)‖2. (2.4)

Let us also recall that the multivalued operator K is said to be upper semicontinuousat x if

xk ∈ X, xk → x

yk ∈ K (xk), yk → y

}⇒ y ∈ K (x).

It is said to be lower semicontinuous at x if

xk ∈ X, xk → x ⇒{

∀y ∈ K (x), ∃{yk} with yk ∈ K (xk)

such that yk → y as k → ∞.

K is continuous at x if it is both upper and lower semicontinuous at x . Moreover,K is continuous on X if and only if it is continuous at every point of X .

Finally, S∗ being contained in the solution set of the quasi-variational inequality(QVI), Assumption A-(a) can be seen as a generalization to the QVIP of the nonempti-ness of the solution set of a variational inequality. Indeed, when K (x) = K for everyx ∈ X , the sets S, S, and K coincide and the set S∗ is the solution set of the VIP. Thisassumption A-(a) has been introduced by Zhang et al. [21].

In order to prove the convergence of the resulting algorithms, we need to use thefollowing result.

Lemma 2.1 ([7], Lemma 4.2) Let x∗ ∈ S∗ and suppose that F is co-coercive on Xwith modulus μ > 1

4 . If zk = PK (xk )(xk − F(xk)), then for any element xk ∈ X, wehave

〈xk − zk, xk − x∗〉 ≥(

1 − 1

4μ

)||xk − zk ||2.

3 The proposed algorithm and its convergence

For solving the QVIP, we consider the following general algorithm.Algorithm QVI

Let l ∈ (0, 1), c ∈ (0, 1), μ > 14 , ρ ≥ 0, and γ ∈ (0, 1).

Set k = 0 and take an arbitrary starting point x0 ∈ X .

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Step 1 Compute zk = PK (xk )(xk − F(xk)). If zk = xk then STOP: xk is a solution to(QVIP). Otherwise, find

yk = (1 − βk)xk + βk zk

where βk = γ lmk and mk is the smallest nonnegative integer m such that

〈F(xk) − F((1 − γ lm)xk + γ lm zk

), xk − zk〉 ≤ c ||xk − zk ||2. (3.1)

Step 2 Choose a direction dk satisfying, for each x∗ ∈ S∗, the inequality

〈βkdk, xk − x∗〉 ≥ ||xk − yk ||2 − βk〈F(xk) − F(yk), xk − yk〉. (3.2)

Compute

dk = ρ

1 + ρ(xk − yk) + 1

1 + ρdk,

xk+1 = PK (xk )(xk − αkβk dk),

where αk > 0.Step 3 Increase k by 1 and go to Step 1.

Before proving the convergence of Algorithm QVI, it remains to define the stepsizeαk and to give some examples of directions dk satisfying (3.2). It is the aim of the nextpropositions.

Proposition 3.1 Let x∗ ∈ S∗ and assume that yk = xk at iteration k. Let also ρ1 =1

1+ρ. Then, −dk is a descent direction at xk for the merit function 1

2 ||x − x∗||2 when

1 − ρ1ρ

4μ− ρ1 c > 0. (3.3)

In particular, this inequality is satisfied when c < 1 + ρ and μ >ρρ1

4(1−ρ1c) .

Proof Using successively the definition of dk , Lemma 2.1, (3.2) and (3.1), we obtain

⟨βk dk, xk − x∗⟩ = βk

⟨ρ1ρ(xk − yk) + ρ1dk, xk − x∗⟩

= ρ1ρβk〈xk − yk, xk − x∗〉 + ρ1βk〈dk, xk − x∗〉≥

(1 − ρ1ρ

4μ

)||xk − yk ||2 − ρ1βk〈F(xk) − F(yk), xk − yk〉 (3.4)

≥(

1 − ρ1ρ

4μ− ρ1 c

)||xk − yk ||2 > 0. (3.5)

But this implies that −dk is a descent direction at xk for the merit function 12 ||x − x∗||2

when (3.3) is satisfied. ��

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Now we can determine the value of αk in Step 2 of Algorithm QVI. Indeed, since〈βk dk, xk − xk〉 = 0, it follows from (3.4) that for

αk =(

1 − ρ1ρ4μ

)||xk − yk ||2 − ρ1βk〈F(xk) − F(yk), xk − yk〉

β2k ||dk ||2

(3.6)

the hyperplane Hk := {x ∈ Rn|〈dk, xk − x〉 = αkβk‖dk‖2} strictly separates xk

from the set S∗. Using the definition of αk and observing that dk is orthogonal to thehyperplane Hk , we obtain that xk − αkβk dk = PHk xk . So, xk+1 is computed thanksto two successive projections: first xk is projected onto Hk (via a closed formula) andafterwards, the resulting vector is projected onto K (xk).

Remark 3.1 Let us observe that when ρ = 0, then ρ1 = 1 and (3.3) becomes c < 1.In that case, it is not necessary to assume that F is co-coercive on X . The monotonicityon F is sufficient to guarantee the convergence of the proposed algorithm.

Now we can give three examples of directions dk satisfying (3.2). (Note that in nextProposition 3.2 and Proposition 3.3 the mapping F needs only to be pseudomonotone).

Proposition 3.2 If d Zk is a direction satisfying (2.3) at iteration k, then the direction

dk = βkd Zk satisfies (3.2). In particular, the direction d1

k := xk − yk + F(yk) satisfies(3.2).

Proof Since xk − yk = βk(xk − zk), we have successively

〈βkdk, xk − x∗〉 = β2k 〈d Z

k , xk − x∗〉≥ β2

k ‖xk − zk‖2 − β2k 〈F(xk) − F(yk), xk − zk〉

= ‖xk − yk‖2 − βk〈F(xk) − F(yk), xk − yk〉.

So the direction dk satisfies (3.2). On the other hand, it was proven in [21] that thedirection d Z

k := xk − zk + 1βk

F(yk) satisfies (2.3). Consequently, the direction d1k ,

being equal to βkd Zk , satisfies (3.2). ��

Proposition 3.3 At iteration k, the two directions

d2k := xk − yk + F(xk) + F(yk)

d3k := xk − yk − βk

(F(xk) − F(yk)

βk

)

introduced in Noor et al. [13] and [14], respectively, satisfy (3.2).

Proof First we observe that

d2k = d1

k + F(xk) and d3k = d1

k − βk F(xk).

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Since the direction d1k satisfies (3.2), it suffices to see that 〈F(xk), xk − x∗〉 ≥ 0

(because F is pseudomonotone) to obtain that the direction d2k satisfies (3.2).

On the other hand, since xk − yk = βk(xk − zk), zk = PK (xk )(xk − F(xk)),x∗ ∈ K (xk) and F is pseudomonotone, we have

〈d3k , xk − x∗〉 = 〈xk − yk + F(yk) − βk F(xk), xk − x∗〉

= βk

⟨xk − zk − F(xk) + F(yk)

βk, xk − x∗

⟩= βk〈xk − F(xk) − zk, xk − x∗〉 + 〈F(yk), xk − x∗〉= βk〈xk − F(xk) − zk, xk − zk〉 + βk〈xk − F(xk) − zk, zk − x∗〉

+ 〈F(yk), xk − yk〉 + 〈F(yk), yk − x∗〉≥ βk〈xk − F(xk) − zk, xk − zk〉 + 〈F(yk), xk − yk〉= βk〈xk − F(xk) − zk, xk − zk〉 + βk〈F(yk), xk − zk〉= βk ||xk − zk ||2 − βk〈F(xk) − F(yk), xk − zk〉.

This implies that

〈βkd3k , xk − x∗〉 ≥ β2

k ||xk − zk ||2 − β2k 〈F(xk) − F(yk), xk − zk〉

= ||xk − yk ||2 − βk〈F(xk) − F(yk), xk − yk〉.

So, the direction d3k satisfies (3.2). ��

Remark 3.2 When ρ = 0, we have that ρ1 = 1 and dk = dk . So, if (3.5) is usedinstead of (3.4), we obtain that for

αk = (1 − c)‖xk − yk‖2

β2k ‖dk‖2

the hyperplane Hk := {x ∈ Rn|〈dk, xk − x〉 = αkβk‖dk‖2} also strictly separates xk

from S∗. With this choice for αk and with dk = d1k , our Algorithm QVI coincides with

Algorithm 2 in [21]. In that case, it is not necessary to assume that F is co-coerciveon X . The monotonicity of F is sufficient to ensure the convergence of the proposedalgorithm.

The following lemma shows that Algorithm QVI is well defined.

Lemma 3.2 Suppose that F is μ-co-coercive on X. At the current iteration k, ifzk = xk , then xk is a solution to QVIP. Otherwise the linesearch condition (3.1) holdsafter finitely many inner iterations.

Proof If zk = xk , then xk = PK (xk )(xk − F(xk)), and thus xk is a solution to QVIP.Next, we suppose, to get a contradiction, that the linesearch condition (3.1) is neversatisfied. Then the following inequality is satisfied for all nonnegative integers m

〈F(xk) − F((1 − γ lm)xk + γ lm zk), xk − zk〉 > c ||xk − zk ||2.

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Using the Cauchy–Schwarz inequality on the left hand side of the last inequality anddividing both sides of the resulting inequality by ||xk − zk ||, we obtain that

||F(xk) − F((1 − γ lm)xk + γ lm zk)|| > c ||xk − zk ||. (3.7)

On the other hand, since F is μ-co-coercive and thus 1/μ-Lipschitz continuous, wehave that

μ||F(xk) − F((1 − γ lm)xk + γ lm zk)|| ≤ γ lm ||xk − zk ||.

Combining this inequality with (3.7), we obtain

γ lm

cμ> 1.

Taking the limit of this inequality as m → ∞, we deduce that 0 ≥ 1, which isimpossible. So, the linesearch condition (3.1) holds after finitely many iterations. ��

Our main result of the paper is to prove the convergence of Algorithm QVI, whichis stated in the following theorems.

Theorem 3.1 Let {xk} be the sequence generated by Algorithm QVI. Suppose thatAssumption A is satisfied and that the parameters ρ, ρ1, μ and c satisfy (3.3). Supposealso that yk = xk for every k and the sequence {dk} is bounded. Then the sequence{xk} generated by Algorithm QVI is bounded, and any limit point of the sequence {xk}is a solution to QVIP.

Precisely because of this result, our aim after the proofs of convergence will be toidentify some concrete search directions dk for which we can guarantee the bounded-ness of the sequences {dk} (see Proposition 3.4 below).

Proof Let x∗ ∈ S∗ and let k ∈ N. Then we have that x∗ ∈ K (xk) and we obtain, usingsuccessively the nonexpansiveness of the projection and (3.4)

||xk+1 − x∗||2 = ||PK (xk )(xk − αkβk dk) − x∗||2≤ ||xk − αkβk dk − x∗||2= ||xk − x∗||2 − 2αk〈βk dk, xk − x∗〉 + α2

k β2k ||dk ||2

≤ ||xk − x∗||2 − 2αk

[(1 − ρ1ρ

4μ

)||xk − yk ||2

− ρ1βk〈F(xk) − F(yk), xk − yk〉]

+ α2k β2

k ||dk ||2.

Consequently, from the definition of αk , we immediately deduce that

||xk+1 − x∗||2 ≤ ||xk − x∗||2

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−((1 − ρ1 ρ

4μ)||xk − yk ||2 − ρ1βk〈F(xk) − F(yk), xk − yk〉

)2

β2k ||dk ||2

. (3.8)

But (3.8) implies that ||xk+1 − x∗|| ≤ ||xk − x∗||. So, the sequence {||xk − x∗||} isconvergent and the sequence {xk} is bounded. Moreover thanks to (3.8), we have

limk→∞

(1 − ρ1 ρ4μ

)||xk − yk ||2 − ρ1βk〈F(xk) − F(yk), xk − yk〉βk ||dk ||

= 0. (3.9)

From (3.5), we obtain easily that

(1 − ρ1 ρ4μ

)||xk − yk ||2 − ρ1βk〈F(xk) − F(yk), xk − yk〉βk ||dk ||

≥ ρ1ρ||xk − yk ||2βk ||dk ||

where ρ = 1 − c + ρ(

1 − 14μ

). Therefore, we have from (3.9) and the definition

of yk that

limk→∞

ρ1 ρβk ||xk − zk ||2||dk ||

= limk→∞

ρ1 ρ||xk − yk ||2βk ||dk ||

= 0. (3.10)

Furthermore, it is easy to verify that {zk} is bounded. Indeed, since x∗ ∈ K (xk),we have successively

||zk || = ||PK (xk )(xk − F(xk))||= ||PK (xk )(xk − F(xk)) + x∗ − PK (xk )(x∗)||≤ ||x∗|| + ||PK (xk )(xk − F(xk)) − PK (xk )(x∗)||≤ ||x∗|| + ||xk − x∗|| + ||F(xk)||.

Since F is continuous and the sequence {xk} is bounded, we can conclude thatthe sequences {zk} and {yk} are also bounded. In addition, since the sequence {dk} isbounded by assumption, we have also that the sequence {dk} is bounded. Therefore itfollows from (3.10) that

limk→∞ βk ||xk − zk ||2 = 0. (3.11)

Let x be a limit point of {xk}. Then there exists a subsequence {xk j } of {xk} con-verging to x when j → ∞. Two cases may occur:

Case 1: inf j βk j = βmin > 0. Then by (3.11), we get lim j→∞ ||xk j − zk j || = 0.Case 2: inf j βk j = βmin = 0. Then there exists a subsequence of {βk j } denoted again{βk j } that converges to 0 as j → ∞. So, for j large enough, βk j = γ lm j with m j > 1.Then γ lm j −1 → 0 and, for j large enough, we can write

〈F(xk j ) − F((1 − γ lm j −1)xk j + γ lm j −1zk j ), xk j − zk j 〉 > c||xk j − zk j ||2.

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Since F is co-coercive, F is also continuous, and using the Cauchy–Schwarzinequality, we obtain that lim j→∞ ||xk j − zk j || = 0.

Therefore, since ||zk j − x || ≤ ||zk j − xk j || + ||xk j − x ||, we obtain in both casesthat zk j → x as j → ∞.

Moreover, by construction of zk , we have that zk ∈ K (xk) for all k. Hence K beingupper semicontinuous on X , we deduce that x ∈ K (x).

On the other hand, since K is lower semicontinuous on X , for any w ∈ K (x),there exists a sequence {wk j } with wk j ∈ K (xk j ), such that wk j → w. Since zk j =PK (xk j )

(xk j − F(xk j )), we obtain that

〈zk j − xk j + F(xk j ), wk j − zk j 〉 ≥ 0,

i.e.,

〈F(xk j ), wk j − zk j 〉 + 〈zk j − xk j , wk j − zk j 〉 ≥ 0.

Taking the limit as j → ∞ gives 〈F(x), w − x〉 ≥ 0 for every w ∈ K (x). But thismeans that x is a solution to QVIP. �

Remark 3.3 One way to obtain that the whole sequence {xk} generated by AlgorithmQVI converges to a solution of QVIP is to impose that every limit point of {xk} belongsto S∗. Indeed, let x be such a limit point. Using (3.8) with x∗ = x , we immediatelydeduce that the sequence {‖xk − x‖} is convergent and thus that the sequence {xk}converges to a solution of QVIP. In the next theorem, we give a condition to assurethat every limit point of {xk} belongs to S∗.

Theorem 3.2 If, in addition to the assumptions of Theorem 3.1, the operator F isstrictly monotone on X, then the sequence {xk} generated by Algorithm QVI is con-vergent to a solution of QVIP.

Proof Let x be a limit point of the sequence {xk}. By Theorem 3.1, x is a solutionto QVIP and by Remark 3.3, we have only to prove that x ∈ S∗ to obtain that {xk}converges to x . In that purpose, let {xk j } be a subsequence of {xk} converging to xand let x∗ ∈ S∗. Then x∗ ∈ K (xk j ) for all j and by the upper semicontinuity of K ,x∗ ∈ K (x). Hence, x being a solution to QVIP, we can write that

〈F(x), x∗ − x〉 ≥ 0. (3.12)

On the other hand, since xk j ∈ K (xk j ) for all j , we have, by definition of S∗, that forall j

〈F(x∗), xk j − x∗〉 ≥ 0.

So, taking the limit as j → ∞, we obtain that

〈F(x∗), x − x∗〉 ≥ 0. (3.13)

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Consequently, from the monotonicity of F , (3.12), and (3.13), we deduce that

〈F(x∗) − F(x), x∗ − x〉 = 0,

which implies that x = x∗ ∈ S∗ because F is strictly monotone on X . ��Proposition 3.4 The directions d1

k , d2k , and d3

k introduced in Propositions 3.2 and 3.3are bounded. So Algorithm QVI is convergent when the sequence of directions {dk} isone of the sequences {d1

k }, {d2k } and {d3

k }.Proof From Theorem 3.2, it is sufficient to prove that each of the sequences of direc-tions {d1

k }, {d2k } and {d3

k } is bounded. In this purpose, first we observe that for all k,xk ∈ K (xk) and that, by the nonexpansiveness of the projection,

||zk − xk || = ||PK (xk )(xk − F(xk)) − PK (xk )(xk)||≤ ||F(xk)||.

This implies that ||yk − xk || = βk ||zk − xk || ≤ βk ||F(xk)||. Therefore, we have, forall k, that

||d1k || = ||xk − yk + F(yk)||

≤ ||xk − yk || + ||F(yk)||≤ βk ||F(xk)|| + ||F(yk)||.

Since F is continuous and the sequences {xk} and {yk} are bounded, we easily deducethat the sequence {d1

k } is bounded. On the other hand, the sequences {F(xk)} and {βk}being bounded, we also obtain that the sequences {d2

k } and {d3k } are bounded. ��

4 Numerical results

In this Section, our aim is to examine the numerical behavior of Algorithm QVIfollowing the choice of directions dk and the value of parameter ρ. Here the directionsdk are successively equal to d1

k , d2k , and d3

k (see Proposition 3.2 and Proposition 3.3),and the values of ρ are equal to 0 or 1. For ρ = 0, the directions dk and dk coincide andAlgorithm QVI is similar to the algorithms proposed in [13,14,21] when the directionsdk are equal to the directions d1

k , d2k , d3

k , respectively. For ρ = 1, the direction dk isdefined by

dk = 1

2(xk − yk) + 1

2dk

and gives rise to three directions d1k , d2

k , d3k corresponding to d1

k , d2k , d3

k , respectively.For the sake of comparison, we have implemented in MATLAB six algorithms cor-responding to two values of parameter ρ and to three directions d1

k , d2k and d3

k . Foreach test example the values of ρ = 0 and ρ = 1 have been considered. As it can

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Table 1 The results of Example 1 for the directions dik , i = 1, 2, 3 and the parameter ρ = 0 and 1

Number of iterations

d1k d2

k d3k

Starting point ρ = 0 ρ = 1 ρ = 0 ρ = 1 ρ = 0 ρ = 1

(0, 0) 393 238 589 322 485 291

(10, 0) / / / / / /

(10, 10) 487 280 572 261 430 260

(0, 10) 308 193 407 233 245 160

(5, 5) 486 278 570 260 427 260

be observed below, the use of directions corresponding to ρ = 1 allows us to get asubstantial improvement in the numerical results (compared with the results obtainedwhen ρ = 0).

Example 1 [3] Consider a two-person game whose QVI formulation involves thefunction F = (F1, F2) and the multivalued mapping K = K1 × K2 defined for eachx = (x1, x2) ∈ R

2 by

F1(x1, x2) = 2x1 + 8

3x2 − 34,

F2(x1, x2) = 2x2 + 5

4x1 − 24.25,

K1(x2) = {y1 ∈ R | 0 ≤ y1 ≤ 10, y1 ≤ 15 − x2},K2(x1) = {y2 ∈ R | 0 ≤ y2 ≤ 10, y2 ≤ 15 − x1}.

The set of solutions is composed of the point (5, 9)T and the line segment joining(9, 6)T to (10, 5)T . The numerical results obtained for this example are listed inTable 1 where different starting points are chosen. The symbol “/” means that thenumber of iterations exceeds 1,000.

Example 2 [3] This is Example 1, except that the set K2(x1) is replaced by

K2(x1) = {y2 ∈ R | 2 ≤ y2 ≤ 10}.

The corresponding numerical results are given in Table 2.Example 3 [21] Consider an oligopolistic market in which five firms supply a

homogeneous product in a noncooperative fashion. The function F = (F1, . . . , F5)

is defined for each x ∈ R5, by

Fi (x) = ci +(

xi

τi

)1/βi

+(

5,000

Q

)1/η (xi

ηQ− 1

), i = 1, . . . , 5

where Q = ∑1≤i≤5 xi . The reader is referred to [21] for more details about this

problem and the different values of the parameters ci , βi , τi , and η. On the other hand,

123



Starting point Number of iterations

d1k d2

k d3k

ρ = 0 ρ = 1 ρ = 0 ρ = 1 ρ = 0 ρ = 1

(0, 0) 486 280 577 257 425 259

(10, 0) 489 284 553 247 427 262

(10, 10) 390 182 464 201 332 186

(0, 10) 308 192 407 230 247 161

(5, 5) 471 272 540 239 413 252


Starting point Number of iterations

d1k d2

k d3k

ρ = 0 ρ = 1 ρ = 0 ρ = 1 ρ = 0 ρ = 1

(10, 10, 10, 10, 10) 135 127 229 181 52 78

(50, 50, 50, 50, 50) 147 139 238 193 60 86

the multivalued mapping K = K1 ×· · ·× K5 is defined for each x ∈ R5 as the product

K1(x−1) × · · · × K5(x−5) where for each i , the set Ki (x−i ) is given by

Ki (x−i ) = {xi ∈ R | 1 ≤ xi ≤ 150, xi +∑j =i

x j ≤ 700 }.

Here we use the notation x−i = (x1, . . . , xi−1, xi+1, . . . , x5) to denote the vector ofcomponents of x except the i-th.

The numerical results obtained for this example are listed in Table 3 for two differentstarting points. The computed solution up to 10−6 is the same for the two starting pointsand identical to the solution given in [21]:

(36.9325, 41.8181, 43.7066, 42.6592, 39.1790).

For each test, the algorithm is terminated at iteration k when ‖xk − zk‖ < ε. In ourcomputational experiments, the following values of the parameters have been used:l = 0.5, c = 0.5, μ = 0.5, γ = 0.99, ε = 10−6. Furthermore, for each test, thenumber of iterations is reported.

From the preliminary results, it seems that the numerical behavior of the algorithmwith the combined direction dk corresponding to ρ = 1 is rather encouraging.

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5 Conclusion

A very general algorithm for solving the class of quasi-variational inequalities hasbeen presented. Six versions of the proposed algorithm have been tested and three ofthem seem to be more efficient thanks to the introduction of a new direction. Howevermany other test problems should be considered and other choices of parameters shouldbe studied to improve the performance of these algorithms. This could be the subjectof future works.

Acknowledgments The authors would like to thank the two anonymous referees for their valuable com-ments and suggestions that improved the original version of the paper substantially. This research is fundedby the Department of Science and Technology at Ho Chi Minh City, Vietnam. Computing resources andsupport provided by the Institute for Computational Science and Technology at Ho Chi Minh City (ICST)are gratefully acknowledged.

References

1. Bensoussan, A.: Points de Nash dans le cas de fonctionnelles quadratiques et jeux différentiels linéairesà N personnes. SIAM J. Control Optim. 12, 460–499 (2005)

2. Facchinei, F., Pang, J.S.: Finite dimensional variational inequalities and complementarity problems.Springer, Berlin (2003)

3. Facchinei, F., Kanzow, C., Sagratella, S.: QVILIB: a library of quasi-variational inequality test prob-lems. Pac. J. Optim. 9, 225–250 (2013)

4. Facchinei, F., Kanzow, C., Sagratella, S.: Solving quasi-variational inequalities via their KKT condi-tions. Math. Program, Ser. A. 44 (2013). doi:10.1007/s10107-013-0637-0

5. Fukushima, M.: A class of gap functions for quasi-variational inequality problems. J. Ind. Manag.Optim. 3, 165–171 (2007)

6. Han, D., Lo, H.: Two new self-adaptive projection methods for variational inequality problems. Comput.Math. Appl. 43, 1529–1537 (2002)

7. Han, D., Zhang, H., Qian, G., Xu, L.: An improved two-step method for solving generalized Nashequilibrium problems. Eur. J. Oper. Res. 216, 613–623 (2012)

8. Harker, P.: Generalized Nash games and quasi-variational inequalities. Eur. J. Oper. Res. 54, 81–94(1991)

9. Kocvara, M., Outrata, J.V.: On a class of quasi-variational inequalities. Optim. Methods Softw. 5,275–295 (1995)

10. Lenzen, F., Becker, F., Lellmann, J., Petra, S., Schnörr, C.: A class of quasi-variational inequalities foradaptive image denoising and decomposition. Comput. Optim. Appl. 54, 371–398 (2013)

11. Nagurney, A.: Network economics: a variational inequality approach. Kluwer Academics Publishers,Dordrecht (1999)

12. Nesterov, Y., Scrimali, L.: Solving strongly monotone variational and quasi-variational inequalities.Discret. Contin. Dyn. Syst. Ser. A 31, 1383–1396 (2011)

13. Noor, M.A., Wang, Y., Xiu, N.: Some new projection methods for variational inequalities. Appl. Math.Comput. 137, 423–435 (2003)

14. Noor, M.A., Wang, Y.J., Xiu, N.: Projection iterative schemes for general variational inequalities.J. Inequal Pure Appl. Math. 3(3) (2002)

15. Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria and multi-leader-follower games. Comput. Manag. Sci. 1, 21–56 (2005)

16. Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria and multi-leader-follower games. Comput. Manag. Sci. 6, 373–375 (2009)

17. Scrimali, L.: Pollution control quasi-equilibrium problems with joint implementation of environmentalprojects. Appl. Math. Lett. 25, 385–392 (2012)

18. Strodiot, J.J., Nguyen, T.T.V., Nguyen, V.H.: A new class of hybrid extragradient algorithms for solvingquasi-equilibrium problems. J. Glob. Optim. 56, 373–397 (2013)

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19. Wang, Y., Xiu, N., Wang, C.: Unified framework of extragradient-type methods for pseudomonotonevariational inequalities. J. Optim. Theory Appl. 111, 641–656 (2001)

20. Wei, J.Y., Smeers, Y.: Spatial oligopolistic electricity models with Cournot generators and regulatedtransmission prices. Oper. Res. 47, 102–112 (1999)

21. Zhang, Z., Qu, B., Xiu, N.: Some projection-like methods for the generalized Nash equilibria. Comput.Optim. Appl. 45, 89–109 (2010)

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a class of hybrid methods for quasi-variational inequalities

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