optimality and duality for a multi-objective programming problem involving generalized d-type-i and...

European Journal of Operational Research 173 (2006) 405–418

www.elsevier.com/locate/ejor

Continuous Optimization

Optimality and duality for a multi-objectiveprogramming problem involving generalized d-type-I

and related n-set functions q

S.K. Mishra a, S.Y. Wang b,d, K.K. Lai c,d,*

a Department of Mathematics, Statistics and Computer Science, College of Basic Sciences and Humanities,

Govind Ballabh Pant University of Agriculture and Technology, Pantnagar 263 145, Indiab Institute of Systems Science, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, China

c Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kongd College of Business Administration, Hunan University, Changsha, China

Received 3 November 2003; accepted 21 February 2005Available online 6 June 2005

Abstract

In this paper, we introduce several generalized convexity for a real-valued set function and establish optimality andduality results for a multi-objective programming problem involving generalized d-type-I and related n-set functions.� 2005 Elsevier B.V. All rights reserved.

Keywords: Multi-objective programming; n-set functions; Optimality; Duality; Generalized convexity

1. Introduction

In this paper, we consider the following multi-objective programming problem involving n-set functions:

0377-2doi:10.

q TheIndia uNatura

* CoKowlo

E-m

ðVPÞ minimize F ðSÞ ¼ ðF 1ðSÞ; . . . ; F pðSÞÞsubject to GjðSÞ 5 0; j 2 M ;

217/$ - see front matter � 2005 Elsevier B.V. All rights reserved.1016/j.ejor.2005.02.062

research was supported by the Department of Science and Technology, Ministry of Science and Technology, Government ofnder SERC Fast Track Scheme for Young Scientists 2001–2002 through grant No. SR/FTP/MS-22/2001 and The Nationall Science Foundation of China.

rresponding author. Address: Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue,on, Hong Kong. Fax: +852 27888563.ail address: [email protected] (K.K. Lai).

mailto:[email protected]

406 S.K. Mishra et al. / European Journal of Operational Research 173 (2006) 405–418

where S = (S1, . . . , Sn) 2 Cn, Cn is the n-fold product of a r-algebra C, Fi, i 2 P = {1,2, . . . , p} and Gj,j 2 M = {1,2, . . . , m} are real-valued functions defined on Cn. Let I0 ¼ fS : S 2 Cn;GðSÞ 5 0g be the setof all feasible solutions to (VP), where G = (G1,G2, . . . , Gm).

The analysis of optimization problems with set functions has been the subject of many papers (see [2–4,6,10–14,19,21–25,32–35]). A formulation of optimization problems with set functions was first given inMorris [19]. Its main results are confined only to set functions of a single set. Corley [6] gave conceptsof a partial derivative and a derivative of a real-valued n-set function. Chou et al. [4,5], Kim et al. [10],Lin [11–14], Preda [21,22], and Preda and Minasian [24,25] studied optimality and duality for optimizationproblems involving vector-valued n-set functions. For details, one can refer to Bector and Singh [2], Hsiaand Lee [7], Kim et al. [10], Lin [11–14], Mazzoleni [16], Preda [22], Rosenmuller and Weidner [28], Tanakaand Maruyama [29] and Zalmai [31–35].

Starting from the method used by Jeyakumar and Mond [8], Mangasarian [15], Mond and Weir [18],Mukherjee [20], Ye [30] and Preda and Minasian [26] defined some new classes of scalar and vector func-tions called d-type-I, d-pseudo-type-I and d-quasi-type-I for a multi-objective programming probleminvolving n-set functions and obtained some interesting results on optimality and Wolfe duality.

Recently, Aghezzaf and Hachimi [1] introduced new classes of generalized type-I vector valued functionswhich are different from those defined in Kaul et al. [9]. For details, see Aghezzaf and Hachimi [1]. In thispaper, we extend the generalized d-type-I vector valued functions of Preda et al. [27] as well as the general-ized type-I functions of Aghezzaf and Hachimi [1] to new generalized d-type-I and related n-set functionsand establish optimality and Mond–Weir type duality results for problem (VP).

2. Definitions and preliminaries

In this section we introduce some notions and definitions. For x = (x1, x2, . . . , xm) andy = (y1, y2, . . . , ym) 2 Rm, we denote x 5 y iff xi 5 yi for all i 2 M; x 6 y iff xi 5 yi for all i 2 M withx 5 y; x < y iff xi < yi for all i 2 M; and x 6< y is the negation of x < y. We denote that x 2 Rm

þ iff x = 0.Let (X,C,l) be a finite atomless measure space with L1(X,C,l) separable, and let d be the pseudo-metric

on Cn defined by dðS; T Þ ¼ ½Pn

k¼1l2ðSkDT kÞ�1=2, where S = (S1, . . . , Sn) and T = (T1, . . . , Tn) 2 Cn and D

denotes the symmetric difference. Thus, (Cn, d) is a pseudo-metric space.For h 2 L1(X,C,l), the integral �S h dl is denoted by hh, ISi, where IS 2 L1(X,C,l) is the indicator

(characteristic) function of S 2 C.A concept of differentiability for a real-valued set function was originally introduced by Morris [19]. Its

n-set counterpart was discussed in Corley [6].A function u : C ! R is said to be differentiable at S0 2 C if there exists Du(S0) 2 L1(X,C,l), called the

derivative of u at S0, and w : C · C ! R such that for each S 2 C,

uðSÞ ¼ uðS0Þ þ hDuðS0Þ; IS � IS0i þ wðS; S0Þ;

where w(S, S0) is o(d(S, S0)), that is, limdðS;S0Þ!0wðS;S0ÞdðS;S0Þ ¼ 0.

A function F : Cn ! R is said to have a partial derivative at S0 ¼ ðS01; . . . ; S0

nÞ with respect to its kth argu-ment if the function

uðSkÞ ¼ F ðS01; . . . ; S0

k�1; S0kþ1; . . . ; S0

nÞ

has derivative DuðS0kÞ and we define DkF ðS0Þ ¼ DuðS0

kÞ. If DkF(S0) exists for 1 5 k 5 n, we denoteDF(S0) = (D1F(S0), . . . , DnF(S0)).

F : Cn ! R is said to be differentiable at S0 ¼ ðS01; . . . ; S0

nÞ if there exist DF(S0) and w : Cn · Cn ! R suchthat

S.K. Mishra et al. / European Journal of Operational Research 173 (2006) 405–418 407

F ðSÞ ¼ F ðS0Þ þXn

k¼1

hDkF ðS0Þ; ISk � IS0ki þ wðS; S0Þ;

where w(S, S0) is o(d(S, S0)).A feasible solution S0 of (VP) is said to be an efficient solution of (VP) if there exists no other feasible

solution S of (VP) such that Fi(S) 5 Fi(S0), for all i 2 M with strict inequality for at least one i 2 M.

A feasible solution S0 of (VP) is said to be a weakly minimum (weakly efficient solution) of (VP) if thereexists no other feasible solution S of (VP) such that Fi(S) < Fi(S

0), for all i 2 M.Let q1,q2, . . . ,qp,q01,q02, . . . ,q0m,q,q 0 be real numbers, and denote �q ¼ ðq1; . . . ; qpÞ and �q0 ¼ ðq01; . . . ; q0mÞ.

Let h : Cn · Cn ! R+ be a function such that h(S, S0) 5 0 for S 5 S0.Along the lines of Aghezzaf and Hachimi [1] and Preda et al. [27], we define the following classes of n-set

functions, called (q,q 0, d)—strong pseudo-quasi-type-I, (q,q 0, d)—weak strictly-pseudo-quasi-type-I,(q,q 0, d)—weak strictly pseudo-type-I, (q,q 0, d)—weak quasi-strictly-pseudo-type-I functions.

Definition 2.1. (F, G) is said to be (q,q 0, d)—weak strictly-pseudo-quasi-type-I at S0 2 Cn with respect toki,dj : Cn · Cn! R+n{0}, i 2 P, j 2 M, if for all S 2 I0, we have
Xp
i¼1

ciðS; S0ÞF iðSÞ 6Xp

i¼1

ciðS; S0ÞF iðS0Þ )Xp

i¼1

Xn

k¼1

hDkF iðS0Þ; ISk � IS0ki < �qhðS; S0Þ

and

�Xm

j¼1

djðS; S0ÞGjðS0Þ 5 0)Xm

j¼1

Xn

k¼1

hDkGjðS0Þ; ISk � IS0ki 5 � q0hðS; S0Þ.

The above definition extends the concept of (q,q 0, d)-pseudo-quasi-type-I functions of Preda et al. [27] aswell as the concept of weak strictly-pseudo-quasi-type-I functions of Aghezzaf and Hachimi [1].

Definition 2.2. (F, G) is said to be (q,q 0, d)-strong-pseudo-quasi-type-I at S0 2 Cn with respect toki,dj : Cn · Cn! R+n{0}, i 2 P, j 2 M, if for all S 2 I0, we have
Xp
i¼1


i¼1


i¼1

Xn

k¼1

hDkF iðS0Þ; ISk � IS0ki 6 �qhðS; S0Þ

and

�Xm

j¼1


j¼1

Xn

k¼1

hDkGjðS0Þ; ISk � IS0ki 5 � q0hðS; S0Þ.

Definition 2.3. (F, G) is said to be (q,q 0, d)-weak quasi-strictly-pseudo-type-I at S0 2 Cn with respect toki,dj : Cn · Cn! R+n{0}, i 2 P, j 2 M, if for all S 2 I0, we have
Xp
i¼1


i¼1


i¼1

Xn

k¼1

hDkF iðS0Þ; ISk � IS0ki 5 � qhðS; S0Þ

and

�Xm

j¼1


j¼1

Xn

k¼1

hDkGjðS0Þ; ISk � IS0ki 6 �q0hðS; S0Þ.


Definition 2.4. (F, G) is said to be (q,q 0, d)-weak strictly pseudo-type-I at S0 2 Cn with respect toki,dj : Cn · Cn ! R+n{0}, i 2 P, j 2 M, if for all S 2 I0, we have
Xp
i¼1


i¼1


i¼1

Xn

k¼1

hDkF iðS0Þ; ISk � IS0ki < �qhðS; S0Þ

and

�Xm

j¼1


j¼1

Xn

k¼1

hDkGjðS0Þ; ISk � IS0ki < �q0hðS; S0Þ.

Remark 2.1. The above definitions are extensions of the corresponding concepts in Aghezzaf and Hachimi[1]. These definitions are also different from the definitions given in Preda [27], Mishra [17] and Kaul et al.[9].

The following lemma will be needed in Section 4 and its proof can be found in Zalmai [33].

Lemma 2.1. Let S0 be an efficient (or weakly efficient) solution for (VP) and let Fi, i 2 P and Gj, j 2 M be

differentiable at S0. Then there exist k 2 Rpþ, l 2 Rm

þ, (k,l) 5 0 such that
Xn
k¼1

Xp

i¼1

kiDkF iðS0Þ þXm

j¼1

ljDkGjðS0Þ; ISk � IS0k

* += 0; for all S 2 Cn

ljGjðS0Þ ¼ 0; j 2 M .

Definition 2.5. A feasible solution S0 is said to be a regular feasible solution if there exists bS 2 Cn such that

GjðS0Þ þXn

k¼1

hDkGjðS0Þ; I Sk� IS0

ki < 0; j 2 M .

3. Optimality conditions

In this section, we establish some sufficient optimality conditions for a weakly efficient solution to (VP)under the assumptions of new types of generalized convexity introduced in Section 2.

Theorem 3.1. Let S0 be a feasible solution for (VP). Suppose that

(A1) there exist k0i = 0, i 2 P,

Ppi¼1k

0i ¼ 1, l0

j = 0, j 2 M0 = M(S0) = {j 2 M : Gj(S0) = 0} such that
Xp
i¼1

Xn

k¼1

k0i hDkF iðS0Þ; ISk � IS0

ki þ

Xj2M0

Xn

k¼1

l0j hDkGjðS0Þ; ISk � IS0

ki = 0; 8S 2 Cn

(A2) (F, G0) is (q,q 0, d)-strong pseudo-quasi-type-I at S0, with respect to ci, and dj;

(A3) q + q 0 = 0.

Then S0 is a weak minimum to (VP).

Proof. Assume that S0 is not a weakly efficient solution to (VP). There is a feasible solution S 5 S0 to (VP)such that

F iðSÞ < F iðS0Þ; for all i 2 P . ð1Þ


According to (A2), there exist ki, dj : Cn · Cn ! R+n{0}, i 2 P, j 2 M0, such that for any S 2 I0,
Xp
i¼1

ciðS; S0Þk0i F iðSÞ <

Xp

i¼1

ciðS; S0Þk0i F iðS0Þ )

Xp

i¼1

Xn

k¼1


ki < �qhðS; S0Þ ð2Þ

and

�Xj2M0

djðS; S0Þl0j GjðS0Þ 5 0)

Xj2M0

Xn

k¼1


ki 5 � q0hðS; S0Þ. ð3Þ

From (1) and k0i = 0, i 2 P, with

Ppi¼1k

0i ¼ 1 and ci > 0, for any i 2 P, we get
Xp
i¼1

ciðS; S0Þk0i F iðSÞ <

Xp

i¼1

ciðS; S0Þk0i F iðS0Þ.

By (2), we get
Xp
i¼1

Xn

k¼1


ki < �qhðS; S0Þ. ð4Þ

Since S0 is feasible to (VP) and Gj(S0) = 0 for j 2 M0, we obtain

�Xj2M0

djðS; S0Þl0j GjðS0Þ 6 0.

This relation together with (3) implies
Xj2M0
Xn

k¼1



By (4) and (5), we get
Xp
i¼1

Xn

k¼1


ki þ

Xj2M0

Xn

k¼1


ki < �ðqþ q0ÞhðS; S0Þ 5 0;

which contradicts (A1). Hence, S0 is a weak minimum of (VP). h


(B1) there exist k0i = 0, i 2 P,

Ppi¼1k

0i ¼ 1, l0

Xp
i¼1

Xn

k¼1


ki þ

Xj2M0

Xn

k¼1


ki = 0; 8S 2 Cn

(B2) ððk0i F Þi2P ; ðl0

j GjÞj2M0Þ is (q,q 0, d)-weak strictly pseudo-quasi-type-I at S0, with respect to ci, and dj;

(B3) q + q 0 = 0.


Proof. Assume that S0 is not a weakly efficient solution to (VP). Thus, there is a feasible solution S 5 S0 to(VP) such that

F iðSÞ < F iðS0Þ; for all i 2 P . ð6Þ


According to (B2), there exist ki,dj : Cn · Cn! R+n{0}, i 2 P, j 2 M0, such that for any S 2 I0,
Xp
i¼1

ciðS; S0Þk0i F iðSÞ 6

Xp

i¼1


Xp

i¼1

Xn

k¼1



and

�Xj2M0


Xj2M0

Xn

k¼1




Ppi¼1k

0i ¼ 1 and ci > 0, for any i 2 P, we get
Xp
i¼1


Xp

i¼1


By (7), we get
Xp
i¼1

Xn

k¼1



Since S0 is feasible to (VP) and Gj(S0) = 0 for j 2 M0, we obtain

�Xj2M0


Xj2M0
Xn

k¼1



Xp
i¼1

Xn

k¼1


ki þ

Xj2M0

Xn

k¼1



which contradicts (B1). Hence, S0 is a weak minimum for (VP). h


(C1) there exist k0i = 0, i 2 P,

Ppi¼1k

0i ¼ 1, l0

Xp
i¼1

Xn

k¼1


ki þ

Xj2M0

Xn

k¼1


ki = 0; 8S 2 Cn

(C2) ððk0i F Þi2P ; ðl0

j GjÞj2M0Þ is (q,q 0, d)-weak strictly pseudo-type-I at S0, with respect to ci, and dj;

(C3) q + q 0 = 0.


Proof. Assume that S0 is not a weakly efficient solution to (VP). Thus, there is a feasible solution S 5 S0 to(VP) such that

F iðSÞ < F iðS0Þ; for any i 2 P . ð11Þ


According to (C2), there exist ki,dj : Cn · Cn ! R+n{0}, i 2 P, j 2 M0, such that for any S 2 I0,
Xp
i¼1


Xp

i¼1


Xp

i¼1

Xn

k¼1



and

�Xj2M0


Xj2M0

Xn

k¼1


ki < �q0hðS; S0Þ. ð13Þ


Ppi¼1k

0i ¼ 1 and ci > 0, for all i 2 P, we get
Xp
i¼1


Xp

i¼1


By (12), we get
Xp
i¼1

Xn

k¼1



Since S0 is a feasible solution of (VP) and Gj(S0) = 0 for j 2 M0, we obtain

�Xj2M0


Xj2M0
Xn

k¼1


ki < �q0hðS; S0Þ. ð15Þ

Xp
i¼1

Xn

k¼1


ki þ

Xj2M0

Xn

k¼1



which contradicts (C1). Hence, S0 is a weak minimum for (VP). h

4. Mond–Weir duality

In this section, we consider the following Mond–Weir dual problem:

ðMDÞ maximize F ðT Þ

subject toXp

i¼1

Xn

k¼1

kihDkðF ÞðT Þ; ISk � IT k i þXm

j¼1

Xn

k¼1

ljhDkðGÞðT Þ; ISk � IT k i = 0;

ljGjðT Þ = 0; j ¼ 1; . . . ;m;

ki = 0; i 2 P andXp

i¼1

ki ¼ 1; lj = 0; j 2 M ;

where S 2 Cn and T 2 Cn.


Let N be the set of all feasible solutions to (MD).

Theorem 4.1 (Weak duality). Suppose that S 2 I0 and (T,k,l) 2 N. Assume that

(D1) (F, G0) is (q,q 0, d)-strong pseudo-quasi-type-I at T with respect to ci, and dj and k > 0;

(D2) q + q 0 = 0.

Then F(S) 6< F(T).

Proof. We proceed by contradiction. If there exist S 2 I0 and (T,k,l) 2 N such that F(S) < F(T). Sinceci > 0, we have
Xp
i¼1

ciðS; T ÞF iðSÞ 6Xp

i¼1

ciðS; T ÞF iðT Þ. ð16Þ

Since (T,k,l) 2 N, it follows that
Xm
j¼1

ljGjðT Þ = 0; that is; �Xm

j¼1

ljGjðT Þ 5 0.

Because dj > 0, we have

�Xm

j¼1

djðS; T ÞljGjðT Þ 5 0. ð17Þ

By (D1), (16) and (17) yield
Xp
i¼1

Xn

k¼1

kihDkF iðT Þ; ISk � IT k i 6 �qhðS; T Þ

and
Xm
j¼1

Xn

k¼1

ljhDkGjðT Þ; ISk � IT k i 5 � q0hðS; T Þ.

Since k > 0, the above two inequalities imply
Xn
k¼1

hDkðkF ÞðT Þ þ DkðlGÞðT Þ; ISk � IT k i < �ðqþ q0ÞhðS; T Þ 5 0;

which contradicts the feasibility of (T,k,l) to (MD). Thus, F(S) 6< F(T). The proof is completed. h

Theorem 4.2 (Weak duality). Suppose that S 2 I0 and (T, k,l) 2 N. Assume that

(E1) ððk0i F Þi2P ; ðl0

j GjÞj2MÞ is (q,q 0, d)-weak strictly pseudo-quasi-type-I at T with respect to ci, and dj;

(E2) q + q 0 = 0.

Then F(S) 6< F(T).

Proof. We proceed by contradiction. If there exist S 2 I0 and (T,k,l) 2 N such that F(S) < F(T). Becauseci > 0, we have
Xp
i¼1


i¼1



Xm
j¼1


j¼1

ljGjðT Þ 5 0.


�Xm

j¼1


By (E1), (18) and (19) yield
Xp
i¼1

Xn

k¼1

kihDkF iðT Þ; ISk � IT k i < �qhðS; T Þ

and
Xm
j¼1

Xn

k¼1

ljhDkGjðT Þ; ISk � IT k i 5 � q0hðS; T Þ.

The above two inequalities imply
Xn
k¼1

hDkðkF ÞðT Þ þ DkðlGÞðT Þ; ISk � IT k i < �ðqþ q0ÞhðS; T Þ 5 0.

This contradicts the feasibility of (T,k,l) for (MD). Thus, F(S) 6< F(T). The proof is completed. h

Theorem 4.3 (Weak duality). Suppose that S 2 I0 and (T, k,l) 2 N. Assume that

(F1) ððk0i F Þi2P ; ðl0

j GjÞj2MÞ is (q,q 0, d)-weak strictly pseudo-type-I at T with respect to ci, and dj;

(F2) q + q 0 = 0.

Then F(S) 6< F(T).

Proof. We proceed by contradiction. If there exist S 2 I0 and (T,k,l) 2 N such that F(S) < F(T). Sinceci > 0, we have
Xp
i¼1


i¼1


Xm
j¼1


j¼1

ljGjðT Þ 5 0.


�Xm

j¼1


By (F1), (20) and (21) yield
Xp
i¼1

Xn

k¼1

kihDkF iðT Þ; ISk � IT k i < �qhðS; T Þ


and
Xm
j¼1

Xn

k¼1

ljhDkGjðT Þ; ISk � IT k i < �q0hðS; T Þ.

The above two inequalities imply
Xn
k¼1

hDkðkF ÞðT Þ þ DkðlGÞðT Þ; ISk � IT k i < �ðqþ q0ÞhðS; T Þ 5 0.

This contradicts the feasibility of (T,k,l) for (MD). Thus, F(S) 6< F(T). The proof is completed. h

Theorem 4.4 (Strong duality). Let S0 2 I0 be a regular weak minimum to (VP). Then there exist k0 2 Rp and

l0 2 Rm such that (S0,k0,l0) is a feasible solution to (MD) and the values of the objective functions of (VP)

and (MD) are equal at S0 and (S0,k0,l0), respectively. Furthermore, if the conditions of Theorem 4.1 (respec-

tively Theorem 4.2 or Theorem 4.3) hold for each feasible (T, k,l) to (MD), then (S0,k0,l0) is a weak max-

imum to (MD).

Proof. By Lemma 2.1, there exist k0i = 0, i 2 P with

Ppi¼1k

0i ¼ 1 and l0

j = 0, j 2 M such that (S0,k0,l0) isfeasible for (MD) and the values of the objective functions of (VP) and (MD) are equal. The last part fol-lows directly from Theorem 4.1 (respectively Theorem 4.2 or Theorem 4.3). h

5. General Mond–Weir duality

In this section, we would like to study general Mond-Weir type duality for (VP) and establish weak andstrong duality theorems under the generalized convexity introduced in Section 2.

Consider the following general Mond–Weir type of dual problem:X
ðGMDÞ maximize F ðT Þ þ
j2J0

ljGjðT Þ

subject toXp

i¼1

Xn

k¼1

kihDkðF ÞðT Þ; ISk � IT k i þXm

j¼1

Xn

k¼1

ljhDkðGÞðT Þ; ISk � IT k i = 0;Xj2Ja

ljGjðT Þ = 0 for 1 5 a 5 r;

k = 0; l = 0 andXp

i¼1

ki ¼ 1;

where S 2 Cn, Ja, 0 5 a 5 r is a partitions of set M with Js \ Jt = /(empty) for s 5 t andSr

s¼0J s ¼ M .

Theorem 5.1 (Weak duality). Suppose that for any S 2 I0 and (T, k,l) feasible for (GMD),

(G1) k > 0 and ððF þP

j2J0ljGjÞð�Þ;

Pj2Ja

ljGjð�ÞÞ is (q,q 0, d)-strong pseudo-quasi-type-I at T with respect to

ci and dj for any a, 1 5 a 5 r;

(G2) q + q 0 = 0.

Then

F ðSÞ¥ F ðT Þ þXj2J0

ljGjðT Þ.


Proof. Suppose to the contrary that the result does not hold. Since S 2 I0 and l = 0, we haveX X
F ðSÞ þ
j2J0

ljGjðSÞ < F ðT Þ þj2J0

ljGjðT Þ.

From the feasibility of (T,k,l), we have

�Xj2Ja

ljGjðT Þ 5 0; for any 0 5 a 5 r.

Since ci > 0 and dj > 0, for j 2 M � J0, from the above two inequalities, we have
Xp
i¼1

ciðS; T Þ F iðSÞ þXj2J0

ljGjðSÞ !

<Xp

i¼1

ciðS; T Þ F iðT Þ þXj2J0

ljGjðT Þ !

ð22Þ

and

�Xj2Ja


By (G1), (22) and (23), we have
Xp
i¼1

Xn

k¼1

ki DkF iðT Þ þXj2J0

ljDkGjðT Þ; ISk � IT k

* +< �qhðS; T Þ ð24Þ

and
Xn
k¼1

Xj2Ja

ljhDkGjðT Þ; ISk � IT k i 5 � q0hðS; T Þ. ð25Þ

Since Ja, 0 5 a 5 r are partitions of set M, k = 0 andPp

i¼1ki ¼ 1, (24) and (25) yield
Xp
i¼1

Xn

k¼1

kihDkðkF ÞðT Þ; ISk � IT k i þXm

j¼1

Xn

k¼1

ljhDkðlGÞðT Þ; ISk � IT k i < �ðqþ q0ÞhðS; T Þ 5 0.

This contradicts the feasibility of (T,k,l) for (GMD). The proof is completed. h


(H1) ððF þP

j2J0ljGjÞð�Þ;

Pj2Ja

ljGjð�ÞÞ is (q,q 0, d)-weak strictly pseudo-quasi-type-I at T with respect to ci

and dj for any a, 1 5 a 5 r;

(H2) q + q 0 = 0.

Then


ljGjðT Þ.

Proof. Suppose to the contrary that the result doe not hold. Since S 2 I0 and l = 0, we have

F ðSÞ þXj2J0

ljGjðSÞ < F ðT Þ þXj2J0

ljGjðT Þ.

From the feasibility of (T,k,l), we have

�Xj2Ja

ljGjðT Þ 5 0; for any 0 5 a 5 r.


Since ci > 0 and dj > 0, for j 2 M � J0, from the above two inequalities, we have

Xp

i¼1

ciðS; T Þ F iðSÞ þXj2J0

ljGjðSÞ !

<Xp

i¼1

ciðS; T Þ F iðT Þ þXj2J0

ljGjðT Þ !

ð26Þ

and

�Xj2Ja

djðS; T ÞljGjðT Þ 5 0; for any 0 5 a 5 r. ð27Þ

By (H1), (26) and (27), we have
Xp
i¼1

Xn

k¼1

ki DkF iðT Þ þXj2J0

ljDkGjðT Þ; ISk � IT k

* +< �qhðS; T Þ ð28Þ

and
Xn
k¼1

Xj2Ja

ljhDkGjðT Þ; ISk � IT k i 5 � q0hðS; T Þ. ð29Þ

Since Ja, 0 5 a 5 r is a partition of set M, k = 0 andPp

i¼1ki ¼ 1, (28) and (29) yield
Xp
i¼1

Xn

k¼1

kihDkðkF ÞðT Þ; ISk � IT k i þXm

j¼1

Xn

k¼1

ljhDkðlGÞðT Þ; ISk � IT k i < �ðqþ q0ÞhðS; T Þ 5 0.

This contradicts the feasibility of (T,k,l) for (GMD). The proof is completed. h


(K1) ððF þP

j2J0ljGjÞð�Þ;

Pj2Ja

ljGjð�ÞÞ is (q,q 0, d)-weak strictly pseudo-type-I at T with respect to ci and dj

for any a, 1 5 a 5 r;

(K2) q + q 0 = 0.

Then


ljGjðT Þ.

Proof. Because ððF þP

j2J0ljGjÞð�Þ;

Pj2Ja

ljGjð�ÞÞ is of (q,q 0, d)-weak strictly pseudo-type-I, we get (28)and

Xn

k¼1

Xj2Ja

ljhDkGjðT Þ; ISk � IT k i < �q0hðS; T Þ. ð30Þ

Adding (28) and (30), we get a contradiction to the feasibility of (T,k,l). This completes the proof. h

Theorem 5.4 (Strong duality). Let S0 2 I0 be a regular weak minimum to (VP). Then there exist k0 2 Rp and

l0 2 Rm such that (S0,k0,l0) is a feasible solution to (GMD) and the values of the objective functions of (VP)

and (GMD) are equal at S0 and (S0,k0,l0), respectively. Furthermore, if the conditions in (respectivelyTheorem 5.2 or Theorem 5.3) hold for each feasible (T,k,l) to (GMD), then (S0,k0,l0) is a weak maximum

to (GMD).


Proof. By Lemma 2.1, there exist k0i = 0, i 2 P with

Ppi¼1k

0i ¼ 1 and l0

j = 0, j 2 M such that (S0,k0,l0) isfeasible for (MD) and the values of the objective functions of (VP) and (MD) are equal. The last part fol-lows directly from Theorem 5.1 (respectively Theorem 5.2 or Theorem 5.3). h

Acknowledgment

The authors are thankful to two anonymous referees for their helpful comment that improved the pre-sentation of the paper.

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