Nondifferentiable multiobjective programming ... - Semantic Scholar
European Journal of Operational Research 160 (2005) 218–226 www.elsevier.com/locate/dsw
Nondifferentiable multiobjective programming under generalized d-univexity S.K. Mishra a, S.Y. Wang b, K.K. Lai
c,*
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, India b 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 Kong Received 28 November 2002; accepted 2 June 2003 Available online 6 November 2003
Abstract In this paper, we are concerned with a nondifferentiable multiobjective programming problem with inequality constraints. We introduce four new classes of generalized convex functions by combining the concepts of weak strictly pseudoinvex, strong pseudoinvex, weak quasi invex, weak pseudoinvex and strong quasi invex functions in Aghezzaf and Hachimi [Numer. Funct. Anal. Optim. 22 (2001) 775], d-invex functions in Antczak [Europ. J. Oper. Res. 137 (2002) 28] and univex functions in Bector et al. [Univex functions and univex nonlinear programming, Proc. Admin. Sci. Assoc. Canada, 1992, p. 115]. By utilizing the new concepts, we derive a Karush–Kuhn–Tucker sufficient optimality condition and establish Mond–Weir type and general Mond–Weir type duality results for the nondifferentiable multiobjective programming problem. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Multiobjective programming; Optimality; Duality; Pareto efficient solution; Generalized d-univexity
1. Introduction Convexity plays a vital role in many aspects of mathematical programming including optimality conditions and duality theorems, see for example Mangasarian [11] and Bazaraa et al. [3]. To relax convexity assumptions imposed on the functions in theorems on optimality and duality, *
Corresponding author. Tel.: +852-278-88563; fax: +852278-88560. E-mail address: [email protected] (K.K. Lai).
various generalized convexity concepts have been proposed. Hanson [7] introduced the class of invex functions. Later, Hanson and Mond [8] defined two new classes of functions called type-I and type-II functions, and sufficient optimality conditions were established by using the concepts. Rueda and Hanson [19] further extended type-I functions to the classes of pseudo-type-I and quasi-type-I functions and obtained sufficient optimality conditions for a nonlinear programming problem involving these classes of functions. Kaul et al. [10] considered a multiple objective nonlinear
0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(03)00439-9
S.K. Mishra et al. / European Journal of Operational Research 160 (2005) 218–226
programming problem involving generalized type-I functions and obtained some results on optimality and duality, where the Wolfe and Mond–Weir duals are considered. Univex functions were introduced and studied by Bector et al. [4]. Rueda et al. [20] obtained optimality and duality results for several mathematical programs by combining the concepts of type-I and univex functions. Mishra [14] considered a multiple objective nonlinear programming problem and obtained a few results on optimality, duality and saddle point of a vector valued Lagrangian by combining the concepts of type-I, pseudo-type-I, quasi-type-I, quasi-pseudo-type-I, pseudo-quasitype-I and univex functions. Aghezzaf and Hachimi [1] introduced new classes of weak strictly pseudoinvex, strong pseudoinvex, weak quasi invex, weak pseudoinvex and strong quasi invex functions. It is known that, despite substituting invexity for convexity, many theoretical problems in differentiable programming can be solved [6,7,9]. But the corresponding conclusions cannot be obtained in nondifferentiable programming with the aid of invexity introduced by Hanson [7] because the existence of a derivative is required in the definition of invexity. There exists a generalization of invexity to locally Lipschitz functions, with derivative replaced by the Clarke generalized gradient [5,12,13,15, 16,18]. However, Antczak [2] used directional derivative, in association with a hypothesis of an invex kind following Ye [23]. The necessary optimality conditions in Antczak [2] are different from those cited in the literature. In the present paper, we consider a nondifferentiable and multiobjective programming problem. A few Karush–Kuhn–Tucker type of sufficient optimality conditions are derived for a (weakly) Pareto efficient solution to the problem involving the new classes of directionally differentiable generalized d-univex functions by combining the concepts of univex functions in Bector et al. [4], weak strictly pseudoinvex, strong pseudoinvex, weak quasi invex, weak pseudoinvex and strong quasi invex functions in Aghezzaf and Hachimi [1] and d-invex functions in Antczak [2]. Furthermore, the Mond–Weir type and general Mond–
219
Weir type of duality results are also obtained in terms of right differentials of the aforesaid functions involved in the multiobjective programming problem. The results in this paper extend many earlier works in the literature. 2. Preliminaries In this section, we extend the concepts of weak strictly pseudoinvex, strong pseudoinvex, weak quasi invex, weak pseudoinvex and strong quasi invex functions introduced in Aghezzaf and Hachimi [1] in the setting of Antczak [2] by combining the concept of univexity given in Bector et al. [4], to weak strictly pseudo-d-univex, strong pseudo-d-univex, weak quasi d-univex, weak pseudo-d-univex and strong quasi d-univex functions and give some preliminaries. Consider the following multiobjective programming problem: ðPÞ
min s:t:
f ðxÞ gðxÞ 5 0; x 2 X;
where f : X ! Rk , g : X ! Rm , X is a nonempty open subset of Rn . Let g : X X ! Rn be a vector function. Through the paper, f 0 ðu; gðx; uÞÞ ¼ limþ k!0
f ðu þ kgðx; uÞÞ f ðuÞ : k
A similar notation is made for g0 ðu; gðx; uÞÞ. Let D ¼ fx 2 X : gðxÞ 5 0g be the set of all the feasible solutions for (P) and denote I ¼ f1; ...; kg, M ¼ f1;2;. ..;mg, J ðxÞ ¼ fj 2 M : gj ðxÞ ¼ 0g and e J ðxÞ ¼ fj 2 M : gj ðxÞ < 0g. It is obvious that J ðxÞ [ e J ðxÞ ¼ M. Throughout this paper, the following convention for vectors in Rn will be followed: x>y
if and only if xi > yi ; i ¼ 1; 2; . . . ; n;
x=y xPy
if and only if xi = yi ; i ¼ 1; 2; . . . ; n; if and only if xi = yi ; i ¼ 1; 2; . . . ; n; but x 6¼ y:
In the following definitions, b0 : X X ½0; 1 ! Rþ , /0 : R ! R and g : X X ! Rn is an n-dimensional vector-valued function.
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Definition 1. f is said to be d-univex with respect to b0 , /0 and g at u 2 X if there exist b0 , /0 and g such that for all x 2 X , b0 ðx; uÞ/0 ½f ðxÞ f ðuÞ = f 0 ðu; gðx; uÞÞ: Definition 2. f is said to be weak strictly pseudod-univex with respect to b0 , /0 and g at u 2 X if there exist b0 , /0 and g such that for all x 2 X , b0 ðx; uÞ/0 ½f ðxÞ f ðuÞ 6 0 ) f 0 ðu; gðx; uÞÞ < 0: Definition 3. f is said to be strong pseudo-d-univex with respect to b0 , /0 and g at u 2 X if there exist b0 , /0 and g such that for all x 2 X , b0 ðx; uÞ/0 ½f ðxÞ f ðuÞ 6 0 ) f 0 ðu; gðx; uÞÞ 6 0: Definition 4. f is said to be weak quasi d-univex with respect to b0 , /0 and g at u 2 X if there exist b0 , /0 and g such that for all x 2 X , b0 ðx; uÞ/0 ½f ðxÞ f ðuÞ 6 0 ) f 0 ðu; gðx; uÞÞ 5 0: Definition 5. f is said to be weak pseudo-d-univex with respect to b0 , /0 and g at u 2 X if there exist b0 , /0 and g such that for all x 2 X , b0 ðx; uÞ/0 ½f ðxÞ f ðuÞ < 0 ) f 0 ðu; gðx; uÞÞ 6 0:
Remark 3. If /0 and /1 are the identity function, the functions defined in the above are extensions of b-invex functions as well as of b-invex directionally differentiable functions. For examples of differentiable weak strictly pseudoinvex, strong pseudoinvex, weak quasi invex, weak pseudoinvex and strong quasi invex functions, one can refer to Aghezzaf and Hachimi [1]. Definition 7. A point x 2 D is said to be a weak Pareto efficient solution for (P) if the relation f ðxÞ ¥ f ðxÞ holds for all x 2 D. Definition 8. A point x 2 D is said to be a locally weak Pareto efficient solution for (P) if there is a neighborhood N ðxÞ around x such that f ðxÞ ¥ f ðxÞ holds for all x 2 N ðxÞ \ D. Definition 9. A function f : X ! R is said to be preinvex with respect to g on X if f ðu þ kgðx; uÞÞ 5 kf ðxÞ þ ð1 kÞf ðuÞ holds for all x; u 2 X and k 2 ½0; 1.
Definition 6. f is said to be strong quasi d-univex with respect to b0 , /0 and g at u 2 X if there exist b0 , /0 and g such that for all x 2 X ,
The following results from Antczak [2] and Weir and Mond [22] will be needed in the sequel.
b0 ðx; uÞ/0 ½f ðxÞ f ðuÞ 5 0 ) f 0 ðu; gðx; uÞÞ 6 0:
Lemma 1. If x is a locally weak Pareto or a weak Pareto efficient solution of (P) and if gj is continuous at x for j 2 e J ðxÞ, then the following system of inequalities
Remark 1. If we take b0 ¼ b1 ¼ 1 and /0 and /1 as the identity functions and if function f is differentiable function, then the above definitions reduce to the definitions given in Aghezzaf and Hachimi [1].
f 0 ðx; gðx; xÞÞ < 0; gJ0 ðxÞ ðx; gðx; xÞÞ < 0 has no solution for x 2 X .
Remark 2. If we take b0 ¼ b1 ¼ 1 and /0 and /1 as the identity functions, the functions defined in the above definitions extend the ones given in Suneja and Srivastava [21] to the directionally differentiable form of the functions given in Aghezzaf and Hachimi [1] and Antczak [2].
Lemma 2. Let S be a nonempty set in Rn and let w : S ! Rp be a preinvex function on S. Then either wðxÞ < 0 has a solution x 2 S, or kT wðxÞ = 0 for all x 2 S, or some k 2 Rmþ , but both are never true at the same time.
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Lemma 3 (Fritz John necessary optimality condition). Let x be a weak Pareto efficient solution for (P). Suppose that gj is continuous for j 2 e J ðxÞ, f and g are directionally differentiable at x with f 0 ðx; gðx; xÞÞ, and gJ0 ðxÞ ðx; gðx; xÞÞ preinvex functions of x on X . Then there exist n 2 Rkþ , l 2 Rmþ such that l) satisfies the following conditions: (x; n; nT f 0 ðx; gðx; xÞÞ þ lT g0 ðx; gðx; xÞÞ = 0; 8x 2 X ;
lT gðxÞ ¼ 0;
gðxÞ 5 0:
Definition 10. The function g is said to satisfy the generalized SlaterÕs constraint qualification at x 2 D if g is d-invex at x, and there exists ~x 2 D such that gj ð~xÞ < 0, j 2 J ðxÞ. Definition 11. Suppose that f : X ! Rk is directionally differentiable at u 2 X . Function f is said to be d-invex at u 2 X with respect to g if for any x 2 X, 0
f ðxÞ f ðuÞ = f ðu; gðx; uÞÞ: Lemma 4 (Karush–Kuhn–Tucker necessary optimality condition). Let x be a weak Pareto efficient solution for (P). Suppose that gj is continuous for j2e J ðxÞ, f and g are directionally differentiable at x with f 0 ðx; gðx; xÞÞ, and gJ0 ðxÞ ðx; gðx; xÞÞ preinvex functions of x on X . Moreover, suppose that g satisfies the general Slater’s constraint qualification at x. Then there exists l 2 Rmþ such that (x; l) satisfies the following conditions: f 0 ðx; gðx; xÞÞ þ lT g0 ðx; gðx; xÞÞ = 0
8x 2 X ;
ð1Þ
lT gðxÞ ¼ 0;
ð2Þ
gðxÞ 5 0:
ð3Þ
221
assume that one of the following conditions is satisfied: (a) f is strong pseudo-d-univex at x with respect to some b0 , /0 and g with b0 > 0, a < 0 ) /0 ðaÞ < 0 and lT g is quasi d-univex with respect to b1 , /1 and g with a ¼ 0 ) /1 ðaÞ = 0; (b) f is weak strictly pseudo-d-univex at x with respect to some b0 , /0 and g with b0 P 0, a < 0 ) /0 ðaÞ 6 0 and lT g is quasi d-univex with respect to b1 , /1 and g with a ¼ 0 ) /1 ðaÞ = 0; a ¼ 0 ) /1 ðaÞ = 0; (c) f is weak strictly pseudo-d-univex at x with respect to some b0 ; /0 and g with b0 P 0, a < 0 ) /0 ðaÞ 6 0 and lT g is strong quasi dunivex with respect to b1 , /1 and g with a ¼ 0 ) /1 ðaÞ = 0. Then x is a weak Pareto efficient solution for (P). Proof. We proceed by contradiction. Assume that x is not a weak Pareto efficient solution of (P). Then there is a feasible solution x of (P) such that fi ðxÞ < fi ðxÞ for any i 2 f1; 2; . . . ; kg: Since b0 > 0 and a < 0 ) /0 ðaÞ < 0, from the above inequality, we get b0 ðx; xÞ/0 ½fi ðxÞ fi ðxÞ < 0: Since b1 = 0 and a ¼ 0 ) /1 ðaÞ = 0, by the feasibility of x and (2), we get lT gðxÞ lT gðxÞ 5 0: b1 ðx; xÞ/1 ½ By the generalized d-univex condition in (a) and the above two inequalities, we get f 0 ðx; gðx; xÞÞ < 0
ð4Þ
and lT g0 ðx; gðx; xÞÞ 5 0:
ð5Þ
By (4) and (5), we get 3. Sufficient optimality condition
f 0 ðx; gðx; xÞÞ þ lT g0 ðx; gðx; xÞÞ < 0;
In this section, we establish a Karush–Kuhn– Tucker sufficient optimality condition.
which contradicts (1). For the proof of part (b), assume that x is not a weak Pareto efficient solution of (P). Then there is a feasible solution x of (P) such that
Theorem 1. Let x be a feasible solution for (P) at which conditions (1)–(3) are satisfied. Moreover,
fi ðxÞ < fi ðxÞ for any i 2 f1; 2; . . . ; kg:
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Since b0 P 0, a < 0 ) /0 ðaÞ 6 0, from the above inequality, we get b0 ðx; xÞ/0 ½fi ðxÞ fi ðxÞ 6 0:
ðMWDÞ max f ðyÞ ¼ ðf1 ðyÞ; f2 ðyÞ; . . . ; fk ðyÞÞ s:t:
Since b1 = 0 and a ¼ 0 ) /1 ðaÞ = 0, by the feasibility of x and (2), we get b1 ðx; xÞ/1 ½ lT gðxÞ lT gðxÞ 5 0:
f 0 ðx; gðx; xÞÞ þ lT g0 ðx; gðx; xÞÞ < 0; again a contradiction to (1). Assume that x is not a weak Pareto efficient solution of (P). Then there is a feasible solution x of (P) such that fi ðxÞ < fi ðxÞ for any i 2 f1; 2; . . . ; kg: Since b0 P 0 and a < 0 ) /0 ðaÞ < 0, from the above inequality, we get b0 ðx; xÞ/0 ½fi ðxÞ fi ðxÞ 6 0: Since b1 = 0 and a ¼ 0 ) /1 ðaÞ = 0, from feasibility of x and (2), we get b1 ðx; xÞ/1 ½ lT gðxÞ lT gðxÞ 5 0: By condition (c) and the above two inequalities, we get f 0 ðx; gðx; xÞÞ < 0 and lT g0 ðx; gðx; xÞÞ 6 0: By these two inequalities, we get f 0 ðx; gðx; xÞÞ þ lT g0 ðx; gðx; xÞÞ < 0; (1).
This
ð6Þ ð7Þ
nT e ¼ 1;
ð8Þ
n2
By the generalized d-univex condition in (b) and the above two inequalities, we get (4) and (5). From (4) and (5), we get
which contradicts proof. h
ðnT f 0 þ lT g0 Þðy; gðx; yÞÞ = 0 for all x 2 D; lj gj ðyÞ = 0; j ¼ 1; . . . ; m;
completes
the
Rkþ ;
l2
Rmþ ;
where e ¼ ð1; 1; . . . ; 1Þ 2 Rk . Denote by W ¼ ðy; n; lÞ 2 X Rk Rm : ðnT f 0 þ lT g0 Þðy; gðx; yÞÞ = 0; lj gj ðyÞ = 0; j ¼ 1; . . . ; m; n 2 Rkþ ; nT e ¼ 1; l 2 Rmþ denote the set of all the feasible solutions of (MWD). We denote by prX W the projection of set W on X . Theorem 2 (Weak duality). Let x and (y; n; l) be a feasible solution for (P) and (MWD), respectively. Moreover, suppose that one of the following conditions holds: (a) f is strong pseudo-d-univex at y on D [ prX W with respect to some b0 , /0 and g with n > 0, b0 > 0, a 6 0 ) /0 ðaÞ 6 0, and lT g is quasi dunivex at y on D [ prX W with respect to b1 , /1 and g with a 5 0 ) /1 ðaÞ 5 0; (b) f is weak strictly pseudo-d-univex at y on D [ prX W with respect to some b0 , /0 and g with b0 P 0, a 6 0 ) /0 ðaÞ 6 0, and lT g is quasi dunivex at y on D [ prX W with respect to b1 , /1 and g with a 5 0 ) /1 ðaÞ 5 0; (c) f is weak strictly pseudo-d-univex at y on D [ prX W with respect to some b0 , /0 and g with b0 P 0, a 6 0 ) /0 ðaÞ 6 0, and lT g is quasi dunivex at y on D [ prX W with respect to b1 , /1 and g with a 5 0 ) /1 ðaÞ 5 0. Then
4. Mond–Weir duality In relation to (P), we consider the following dual problem which is in the form of Mond–Weir [17]:
f ðxÞif ðyÞ:
Proof. We proceed by contradiction. Assume that f ðxÞ 6 f ðyÞ:
S.K. Mishra et al. / European Journal of Operational Research 160 (2005) 218–226
Since b0 > 0, a 6 0 ) /0 ðaÞ 6 0, from the above inequality, we get b0 ðx; yÞ/0 ½f ðxÞ f ðyÞ 6 0:
ð9Þ
Since x is feasible for (P) and (y; n; l) is feasible for (MWD), it follows that m m X X lj gj ðxÞ lj gj ðyÞ 5 0: j¼1
j¼1
Since b1 = 0, a 5 0 ) /1 ðaÞ 5 0, from the above inequality, we get " # m m X X b1 ðx; yÞ/1 lj gj ðxÞ lj gj ðyÞ 5 0: ð10Þ j¼1
j¼1
By the generalized d-univex condition in (a), (9) and (10) imply ð11Þ
and lj gj0 ðy; gðx; yÞÞ 5 0:
ð12Þ
j¼1
Since n > 0, from (11) and (12), we get k X
ni fi0 ðy; gðx; yÞÞ þ
i¼1
m X
lj gj0 ðy; gðx; yÞÞ < 0;
ð13Þ
l¼1
which contradicts (6). For the proof of part (b), again assume that f ðxÞ 6 f ðyÞ: Since b0 P 0, a 6 0 ) /0 ðaÞ 6 0, from the above inequality, we get b0 ðx; yÞ/0 ½f ðxÞ f ðyÞ 6 0:
ð14Þ
Since x is feasible for (P) and (y; n; l) is feasible for (MWD), it follows that m m X X lj gj ðxÞ lj gj ðyÞ 5 0: j¼1
f 0 ðy; gðx; yÞÞ < 0; and m X
lj gj0 ðy; gðx; yÞÞ 5 0:
j¼1
Since n = 0, the above two inequalities imply (13), again a contradiction to (6). For the proof of part (c), proceeding as in part (b), we get (14) and (15). By the generalized univexity condition in part (c), (14) and (15) imply f 0 ðy; gðx; yÞÞ < 0;
ð16Þ
and m X lj gj0 ðy; gðx; yÞÞ 6 0:
ð17Þ
j¼1
f 0 ðy; gðx; yÞÞ 6 0; m X
223
Since n = 0, (16) and (17) imply (13), again a contradiction to (6). This completes the proof. h Theorem 3 (Strong duality). Let x be a locally weak Pareto efficient solution or a weak Pareto efficient solution for (P) at which the generalized Slater’s constraint qualification is satisfied. Let (f ; g) be directionally differentiable at x with f 0 ðx; gðx; xÞÞ, and g0 ðx; gðx; xÞÞ preinvex functions on X and gj be continuous for j 2 b J ðxÞ. Then there exists l 2 Rmþ such that (x; 1; l) is feasible for (MWD). If the weak duality between (P) and (MWD) in Theorem 2 holds, then (x; 1; l) is a locally weak Pareto efficient solution for (MWD). Proof. Since x satisfies all the conditions of Lemma 4, there exists l 2 Rmþ such that conditions (1)–(3) hold. By (1)–(3), we have that (x; 1; l) is feasible for (MWD). By the weak duality, it follows that (x; 1; l) is a locally weak Pareto efficient solution for (MWD). h
j¼1
Since b1 = 0, a 5 0 ) /1 ðaÞ 5 0, from the above inequality, we get " # m m X X b1 ðx; yÞ/1 lj gj ðxÞ lj gj ðyÞ 5 0: ð15Þ j¼1
j¼1
By the generalized d-univex condition in (b), (14) and (15) imply
l) be a Theorem 4 (Converse duality). Let (y ; n; weak Pareto efficient solution for (MWD). If the hypothesis of Theorem 2 holds at y on D [ prX W , then y is a weak Pareto efficient solution for (P). Proof. We proceed by contradiction. Assume that y is not a weak Pareto efficient solution for (P), that is, there exists ~x 2 D such that
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f ð~xÞ < f ðy Þ: We know from condition (a) of Theorem 2 that b0 > 0, and a < 0 ) /0 ðaÞ < 0, from this and the above inequality, we get
Since ni = 0, the above two inequalities imply (20), again a contradiction to (6). This completes the proof. h
b0 ð~x; yÞ/0 ½f ð~xÞ f ðy Þ < 0:
5. General Mond–Weir duality
By the generalized univexity condition (a) in Theorem 2, we get
We shall continue our discussion on duality for (P) in the present section by considering a general Mond–Weir type dual problem and proving weak and strong duality theorems under an assumption of the generalized d-univexity introduced in Section 2. We consider the following general Mond–Weir type dual to (P):
k X
ni fi0 ðy ; gð~x; y ÞÞ < 0:
ð18Þ
i¼1
l) for From the feasibility of ~x for (P), (y ; n; (MWD), b1 = 0 and a 5 0 ) /1 ðaÞ 5 0, we have " # m m X X b1 ð~x; y Þ/1 lj gj ð~xÞ lj gj ðy Þ 5 0: j¼1
j¼1
The above inequality in light of the generalized dunivexity condition (a) in Theorem 2 yields m X ð19Þ lj gj0 ðy ; gð~x; y ÞÞ 5 0: j¼1
ni fi0 ðy ; gð~x; y ÞÞ þ
i¼1
m X
lj gj0 ðy ; gð~x; y ÞÞ < 0:
ð20Þ
j¼1
fi0 ðy ; gð~x; y ÞÞ
0, and (f þ lJ0 gJ0 ; lJt gJt ) is strong pseudoquasi d-type-I univex at y on D [ prX W with respect to some b0 ; b1 ; /0 ; /1 and g with b0 > 0, n > 0, a 6 0 ) /0 ðaÞ 6 0, and a 5 0 ) /1 ðaÞ 5 0 for any t, 1 5 t 5 r. (b) (f þ lJ0 gJ0 ; lJt gJt ) is weak strictly pseudoquasi d-type-I univex at y on D [ prX W with respect
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to some b0 , b1 , /0 , /1 and g with b0 P 0, a 6 0 ) /0 ðaÞ 6 0, and a 5 0 ) /1 ðaÞ 5 0 for any t, 1 5 t 5 r. (c) (f þ lJ0 gJ0 ; lJt gJt ) is weak strictly pseudod-type-I univex at y on D [ prX W with respect to some b0 ; b1 ; /0 , /1 and g with b0 P 0, a 6 0 ) /0 ðaÞ 6 0, and a 5 0 ) /1 ðaÞ 5 0 for any t, 1 5 t 5 r. Then the following cannot hold: f ðxÞ 6 /ðy; n; lÞ:
Similarly, by condition (b), we have 0
ðf þ lJ0 gJ0 0 eÞðy; gðx; yÞÞ < 0 and lJt gJ0 t ðy; gðx; yÞÞ 5 0 81 5 t 5 r: Since n = 0, the above two inequalities yield ! r X T 0 0 n f þ lJt gJt ðy; gðx; yÞÞ < 0: t¼0
Proof. We proceed by contradiction. Suppose that f ðxÞ 6 /ðy; n; lÞ:
ð24Þ
The above inequality leads to (28), which contradicts (21). By condition (c), we get ðf 0 þ lJ0 gJ0 0 eÞðy; gðx; yÞÞ < 0
Since x is feasible for (P) and l = 0, (24) implies that
and
f ðxÞ þ lTJ0 gJ0 ðxÞe 6 f ðyÞ þ lTJ0 gJ0 ðyÞe:
lJt gJ0 t ðy; gðx; yÞÞ < 0
Since b0 > 0, a 6 0 ) /0 ðaÞ 6 0, from the above inequality, we get h i b0 ðx;yÞ/0 f ðxÞ þ lTJ0 gJ0 ðxÞe f ðyÞ þ lTJ0 gJ0 ðyÞe 6 0:
Since n = 0, the above two inequalities yield ! r X T 0 0 lJt gJt ðy; gðx; yÞÞ < 0: n f þ
ð25Þ From the feasibility of x for (P) and (22), we have lTJt gJt ðxÞ lTJt gJt ðyÞ 5 0 for all 1 5 t 5 r: Since b1 = 0, a 5 0 ) /1 ðaÞ 5 0, from the above inequality, we get
b1 ðx;yÞ/1 lTJt gJt ðxÞ lTJt gJt ðyÞ 5 0 for all 1 5 t 5 r: ð26Þ By condition (a), from (25) and (26), we have ðf 0 þ lJ0 gJ0 0 eÞðy; gðx; yÞÞ 6 0 and lJt gJ0 t ðy; gðx; yÞÞ 5 0
225
81 5 t 5 r:
Since n > 0, the above two inequalities yield ! r X T 0 0 n f þ lJt gJt ðy; gðx; yÞÞ < 0: ð27Þ
81 5 t 5 r:
t¼0
The above inequality leads to (28), which contradicts (21). This completes the proof. h Theorem 6 (Strong duality). Let x be a locally weak Pareto efficient solution or a weak Pareto efficient solution for (P) at which the generalized Slater’s constraint qualification is satisfied. Let f and g be directionally differentiable at x with f 0 ðx; gðx; xÞÞ and g0 ðx; gðx; xÞÞ preinvex functions on X and gj be continuous for j 2 b J ðxÞ. Then, there exists l 2 Rmþ such that (x; 1; l) is feasible for (GMWD). Moreover, if the weak duality between (P) and (GMWD) in Theorem 5 holds, then (x; 1; l) is a locally weak Pareto efficient solution for (GMWD). Proof. The proof of this theorem is similar to the proof of Theorem 3. h
t¼0
Since J0 ; . . . ; Jr are partitions of M, (27) is equivalent to ðnT f 0 þ lT g0 Þðy; gðx; yÞÞ < 0; which contradicts the dual constraint (21).
ð28Þ
Acknowledgements The research was supported by the University Grants Commission of India, the National
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Natural Science Foundation of China and the Research Grants Council of Hong Kong.
References [1] B. Aghezzaf, M. Hachimi, Sufficient optimality conditions and duality in multiobjective optimization involving generalized convexity, Numerical Functional Analysis and Optimization 22 (2001) 775–788. [2] T. Antczak, Multiobjective programming under d-invexity, European Journal of Operational Research 137 (2002) 28– 36. [3] M.S. Bazaraa, H.D. Sherali, C.M. Shetty, Nonlinear Programming: Theory and Algorithms, Wiley, New York, 1991. [4] C.R. Bector, S.K. Suneja, S. Gupta, Univex functions and univex nonlinear programming, in: Proceedings of the Administrative Sciences Association of Canada, 1992, pp. 115–124. [5] B.D. Craven, Nondifferentiable optimization by smooth approximation, Optimization 17 (1986) 3–17. [6] R.R. Egudo, M.A. Hanson, Multi-objective duality with invexity, Journal of Mathematical Analysis and its Applications 126 (1987) 469–477. [7] M.A. Hanson, On sufficiency of the Kuhn–Tucker conditions, Journal of Mathematical Analysis and its Applications 80 (1981) 545–550. [8] M.A. Hanson, B. Mond, Necessary and sufficient conditions in constrained optimization, Mathematical Programming 37 (1987) 51–58. [9] V. Jeyakumar, B. Mond, On generalized convex mathematical programming, Journal of Australian Mathematical Society, Series B 34 (1992) 43–53. [10] R.N. Kaul, S.K. Suneja, M.K. Srivastava, Optimality criteria and duality in multiple objective optimization involving generalized invexity, Journal of Optimization Theory and its Applications 80 (1994) 465–482.
[11] O.L. Mangasarian, Nonlinear Programming, McGraw Hill, New York, 1969. [12] S.K. Mishra, Lagrange multipliers saddle points and scalarizations in composite multiobjective nonsmooth programming, Optimization 38 (1996) 93–105. [13] S.K. Mishra, On sufficiency and duality in nonsmooth multiobjective programming, Opsearch 34 (1997) 221–231. [14] S.K. Mishra, On multiple objective optimization with generalized univexity, Journal of Mathematical Analysis and its Applications 224 (1998) 131–148. [15] S.K. Mishra, G. Giorgi, Optimality and duality with generalized semi-univexity, Opsearch 37 (2000) 340–350. [16] S.K. Mishra, R.N. Mukherjee, On generalized convex multiobjective nonsmooth programming, Journal of Australian Mathematical Society, Series B 38 (1996) 140– 148. [17] B. Mond, T. Weir, Generalized concavity and duality, in: S. Schaible, W.T. Ziemba (Eds.), Generalized Concavity Optimization and Economics, Academic Press, New York, 1981, pp. 263–280. [18] T.W. Reiland, Nonsmooth invexity, Bulletin of the Australian Mathematical Society 42 (1990) 437–446. [19] N.G. Rueda, M.A. Hanson, Optimality criteria in mathematical programming involving generalized invexity, Journal of Mathematical Analysis and its Applications 130 (1988) 375–385. [20] N.G. Rueda, M.A. Hanson, C. Singh, Optimality and duality with generalized convexity, Journal of Optimization Theory and its Application 86 (1995) 491–500. [21] S.K. Suneja, M.K. Srivastava, Optimality and duality in nondifferentiable multiobjective optimization involving dtype I and related functions, Journal of Mathematical Analysis and its Applications 206 (1997) 465–479. [22] T. Weir, B. Mond, Pre-invex functions in multiple objective optimization, Journal of Mathematical Analysis and its Applications 136 (1988) 29–38. [23] Y.L. Ye, d-Invexity and optimality conditions, Journal of Mathematical Analysis and its Applications 162 (1991) 242–249.
Nondifferentiable multiobjective programming under generalized d-univexity S.K. Mishra a, S.Y. Wang b, K.K. Lai
c,*
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, India b 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 Kong Received 28 November 2002; accepted 2 June 2003 Available online 6 November 2003
Abstract In this paper, we are concerned with a nondifferentiable multiobjective programming problem with inequality constraints. We introduce four new classes of generalized convex functions by combining the concepts of weak strictly pseudoinvex, strong pseudoinvex, weak quasi invex, weak pseudoinvex and strong quasi invex functions in Aghezzaf and Hachimi [Numer. Funct. Anal. Optim. 22 (2001) 775], d-invex functions in Antczak [Europ. J. Oper. Res. 137 (2002) 28] and univex functions in Bector et al. [Univex functions and univex nonlinear programming, Proc. Admin. Sci. Assoc. Canada, 1992, p. 115]. By utilizing the new concepts, we derive a Karush–Kuhn–Tucker sufficient optimality condition and establish Mond–Weir type and general Mond–Weir type duality results for the nondifferentiable multiobjective programming problem. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Multiobjective programming; Optimality; Duality; Pareto efficient solution; Generalized d-univexity
1. Introduction Convexity plays a vital role in many aspects of mathematical programming including optimality conditions and duality theorems, see for example Mangasarian [11] and Bazaraa et al. [3]. To relax convexity assumptions imposed on the functions in theorems on optimality and duality, *
Corresponding author. Tel.: +852-278-88563; fax: +852278-88560. E-mail address: [email protected] (K.K. Lai).
various generalized convexity concepts have been proposed. Hanson [7] introduced the class of invex functions. Later, Hanson and Mond [8] defined two new classes of functions called type-I and type-II functions, and sufficient optimality conditions were established by using the concepts. Rueda and Hanson [19] further extended type-I functions to the classes of pseudo-type-I and quasi-type-I functions and obtained sufficient optimality conditions for a nonlinear programming problem involving these classes of functions. Kaul et al. [10] considered a multiple objective nonlinear
0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(03)00439-9
S.K. Mishra et al. / European Journal of Operational Research 160 (2005) 218–226
programming problem involving generalized type-I functions and obtained some results on optimality and duality, where the Wolfe and Mond–Weir duals are considered. Univex functions were introduced and studied by Bector et al. [4]. Rueda et al. [20] obtained optimality and duality results for several mathematical programs by combining the concepts of type-I and univex functions. Mishra [14] considered a multiple objective nonlinear programming problem and obtained a few results on optimality, duality and saddle point of a vector valued Lagrangian by combining the concepts of type-I, pseudo-type-I, quasi-type-I, quasi-pseudo-type-I, pseudo-quasitype-I and univex functions. Aghezzaf and Hachimi [1] introduced new classes of weak strictly pseudoinvex, strong pseudoinvex, weak quasi invex, weak pseudoinvex and strong quasi invex functions. It is known that, despite substituting invexity for convexity, many theoretical problems in differentiable programming can be solved [6,7,9]. But the corresponding conclusions cannot be obtained in nondifferentiable programming with the aid of invexity introduced by Hanson [7] because the existence of a derivative is required in the definition of invexity. There exists a generalization of invexity to locally Lipschitz functions, with derivative replaced by the Clarke generalized gradient [5,12,13,15, 16,18]. However, Antczak [2] used directional derivative, in association with a hypothesis of an invex kind following Ye [23]. The necessary optimality conditions in Antczak [2] are different from those cited in the literature. In the present paper, we consider a nondifferentiable and multiobjective programming problem. A few Karush–Kuhn–Tucker type of sufficient optimality conditions are derived for a (weakly) Pareto efficient solution to the problem involving the new classes of directionally differentiable generalized d-univex functions by combining the concepts of univex functions in Bector et al. [4], weak strictly pseudoinvex, strong pseudoinvex, weak quasi invex, weak pseudoinvex and strong quasi invex functions in Aghezzaf and Hachimi [1] and d-invex functions in Antczak [2]. Furthermore, the Mond–Weir type and general Mond–
219
Weir type of duality results are also obtained in terms of right differentials of the aforesaid functions involved in the multiobjective programming problem. The results in this paper extend many earlier works in the literature. 2. Preliminaries In this section, we extend the concepts of weak strictly pseudoinvex, strong pseudoinvex, weak quasi invex, weak pseudoinvex and strong quasi invex functions introduced in Aghezzaf and Hachimi [1] in the setting of Antczak [2] by combining the concept of univexity given in Bector et al. [4], to weak strictly pseudo-d-univex, strong pseudo-d-univex, weak quasi d-univex, weak pseudo-d-univex and strong quasi d-univex functions and give some preliminaries. Consider the following multiobjective programming problem: ðPÞ
min s:t:
f ðxÞ gðxÞ 5 0; x 2 X;
where f : X ! Rk , g : X ! Rm , X is a nonempty open subset of Rn . Let g : X X ! Rn be a vector function. Through the paper, f 0 ðu; gðx; uÞÞ ¼ limþ k!0
f ðu þ kgðx; uÞÞ f ðuÞ : k
A similar notation is made for g0 ðu; gðx; uÞÞ. Let D ¼ fx 2 X : gðxÞ 5 0g be the set of all the feasible solutions for (P) and denote I ¼ f1; ...; kg, M ¼ f1;2;. ..;mg, J ðxÞ ¼ fj 2 M : gj ðxÞ ¼ 0g and e J ðxÞ ¼ fj 2 M : gj ðxÞ < 0g. It is obvious that J ðxÞ [ e J ðxÞ ¼ M. Throughout this paper, the following convention for vectors in Rn will be followed: x>y
if and only if xi > yi ; i ¼ 1; 2; . . . ; n;
x=y xPy
if and only if xi = yi ; i ¼ 1; 2; . . . ; n; if and only if xi = yi ; i ¼ 1; 2; . . . ; n; but x 6¼ y:
In the following definitions, b0 : X X ½0; 1 ! Rþ , /0 : R ! R and g : X X ! Rn is an n-dimensional vector-valued function.
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Definition 1. f is said to be d-univex with respect to b0 , /0 and g at u 2 X if there exist b0 , /0 and g such that for all x 2 X , b0 ðx; uÞ/0 ½f ðxÞ f ðuÞ = f 0 ðu; gðx; uÞÞ: Definition 2. f is said to be weak strictly pseudod-univex with respect to b0 , /0 and g at u 2 X if there exist b0 , /0 and g such that for all x 2 X , b0 ðx; uÞ/0 ½f ðxÞ f ðuÞ 6 0 ) f 0 ðu; gðx; uÞÞ < 0: Definition 3. f is said to be strong pseudo-d-univex with respect to b0 , /0 and g at u 2 X if there exist b0 , /0 and g such that for all x 2 X , b0 ðx; uÞ/0 ½f ðxÞ f ðuÞ 6 0 ) f 0 ðu; gðx; uÞÞ 6 0: Definition 4. f is said to be weak quasi d-univex with respect to b0 , /0 and g at u 2 X if there exist b0 , /0 and g such that for all x 2 X , b0 ðx; uÞ/0 ½f ðxÞ f ðuÞ 6 0 ) f 0 ðu; gðx; uÞÞ 5 0: Definition 5. f is said to be weak pseudo-d-univex with respect to b0 , /0 and g at u 2 X if there exist b0 , /0 and g such that for all x 2 X , b0 ðx; uÞ/0 ½f ðxÞ f ðuÞ < 0 ) f 0 ðu; gðx; uÞÞ 6 0:
Remark 3. If /0 and /1 are the identity function, the functions defined in the above are extensions of b-invex functions as well as of b-invex directionally differentiable functions. For examples of differentiable weak strictly pseudoinvex, strong pseudoinvex, weak quasi invex, weak pseudoinvex and strong quasi invex functions, one can refer to Aghezzaf and Hachimi [1]. Definition 7. A point x 2 D is said to be a weak Pareto efficient solution for (P) if the relation f ðxÞ ¥ f ðxÞ holds for all x 2 D. Definition 8. A point x 2 D is said to be a locally weak Pareto efficient solution for (P) if there is a neighborhood N ðxÞ around x such that f ðxÞ ¥ f ðxÞ holds for all x 2 N ðxÞ \ D. Definition 9. A function f : X ! R is said to be preinvex with respect to g on X if f ðu þ kgðx; uÞÞ 5 kf ðxÞ þ ð1 kÞf ðuÞ holds for all x; u 2 X and k 2 ½0; 1.
Definition 6. f is said to be strong quasi d-univex with respect to b0 , /0 and g at u 2 X if there exist b0 , /0 and g such that for all x 2 X ,
The following results from Antczak [2] and Weir and Mond [22] will be needed in the sequel.
b0 ðx; uÞ/0 ½f ðxÞ f ðuÞ 5 0 ) f 0 ðu; gðx; uÞÞ 6 0:
Lemma 1. If x is a locally weak Pareto or a weak Pareto efficient solution of (P) and if gj is continuous at x for j 2 e J ðxÞ, then the following system of inequalities
Remark 1. If we take b0 ¼ b1 ¼ 1 and /0 and /1 as the identity functions and if function f is differentiable function, then the above definitions reduce to the definitions given in Aghezzaf and Hachimi [1].
f 0 ðx; gðx; xÞÞ < 0; gJ0 ðxÞ ðx; gðx; xÞÞ < 0 has no solution for x 2 X .
Remark 2. If we take b0 ¼ b1 ¼ 1 and /0 and /1 as the identity functions, the functions defined in the above definitions extend the ones given in Suneja and Srivastava [21] to the directionally differentiable form of the functions given in Aghezzaf and Hachimi [1] and Antczak [2].
Lemma 2. Let S be a nonempty set in Rn and let w : S ! Rp be a preinvex function on S. Then either wðxÞ < 0 has a solution x 2 S, or kT wðxÞ = 0 for all x 2 S, or some k 2 Rmþ , but both are never true at the same time.
S.K. Mishra et al. / European Journal of Operational Research 160 (2005) 218–226
Lemma 3 (Fritz John necessary optimality condition). Let x be a weak Pareto efficient solution for (P). Suppose that gj is continuous for j 2 e J ðxÞ, f and g are directionally differentiable at x with f 0 ðx; gðx; xÞÞ, and gJ0 ðxÞ ðx; gðx; xÞÞ preinvex functions of x on X . Then there exist n 2 Rkþ , l 2 Rmþ such that l) satisfies the following conditions: (x; n; nT f 0 ðx; gðx; xÞÞ þ lT g0 ðx; gðx; xÞÞ = 0; 8x 2 X ;
lT gðxÞ ¼ 0;
gðxÞ 5 0:
Definition 10. The function g is said to satisfy the generalized SlaterÕs constraint qualification at x 2 D if g is d-invex at x, and there exists ~x 2 D such that gj ð~xÞ < 0, j 2 J ðxÞ. Definition 11. Suppose that f : X ! Rk is directionally differentiable at u 2 X . Function f is said to be d-invex at u 2 X with respect to g if for any x 2 X, 0
f ðxÞ f ðuÞ = f ðu; gðx; uÞÞ: Lemma 4 (Karush–Kuhn–Tucker necessary optimality condition). Let x be a weak Pareto efficient solution for (P). Suppose that gj is continuous for j2e J ðxÞ, f and g are directionally differentiable at x with f 0 ðx; gðx; xÞÞ, and gJ0 ðxÞ ðx; gðx; xÞÞ preinvex functions of x on X . Moreover, suppose that g satisfies the general Slater’s constraint qualification at x. Then there exists l 2 Rmþ such that (x; l) satisfies the following conditions: f 0 ðx; gðx; xÞÞ þ lT g0 ðx; gðx; xÞÞ = 0
8x 2 X ;
ð1Þ
lT gðxÞ ¼ 0;
ð2Þ
gðxÞ 5 0:
ð3Þ
221
assume that one of the following conditions is satisfied: (a) f is strong pseudo-d-univex at x with respect to some b0 , /0 and g with b0 > 0, a < 0 ) /0 ðaÞ < 0 and lT g is quasi d-univex with respect to b1 , /1 and g with a ¼ 0 ) /1 ðaÞ = 0; (b) f is weak strictly pseudo-d-univex at x with respect to some b0 , /0 and g with b0 P 0, a < 0 ) /0 ðaÞ 6 0 and lT g is quasi d-univex with respect to b1 , /1 and g with a ¼ 0 ) /1 ðaÞ = 0; a ¼ 0 ) /1 ðaÞ = 0; (c) f is weak strictly pseudo-d-univex at x with respect to some b0 ; /0 and g with b0 P 0, a < 0 ) /0 ðaÞ 6 0 and lT g is strong quasi dunivex with respect to b1 , /1 and g with a ¼ 0 ) /1 ðaÞ = 0. Then x is a weak Pareto efficient solution for (P). Proof. We proceed by contradiction. Assume that x is not a weak Pareto efficient solution of (P). Then there is a feasible solution x of (P) such that fi ðxÞ < fi ðxÞ for any i 2 f1; 2; . . . ; kg: Since b0 > 0 and a < 0 ) /0 ðaÞ < 0, from the above inequality, we get b0 ðx; xÞ/0 ½fi ðxÞ fi ðxÞ < 0: Since b1 = 0 and a ¼ 0 ) /1 ðaÞ = 0, by the feasibility of x and (2), we get lT gðxÞ lT gðxÞ 5 0: b1 ðx; xÞ/1 ½ By the generalized d-univex condition in (a) and the above two inequalities, we get f 0 ðx; gðx; xÞÞ < 0
ð4Þ
and lT g0 ðx; gðx; xÞÞ 5 0:
ð5Þ
By (4) and (5), we get 3. Sufficient optimality condition
f 0 ðx; gðx; xÞÞ þ lT g0 ðx; gðx; xÞÞ < 0;
In this section, we establish a Karush–Kuhn– Tucker sufficient optimality condition.
which contradicts (1). For the proof of part (b), assume that x is not a weak Pareto efficient solution of (P). Then there is a feasible solution x of (P) such that
Theorem 1. Let x be a feasible solution for (P) at which conditions (1)–(3) are satisfied. Moreover,
fi ðxÞ < fi ðxÞ for any i 2 f1; 2; . . . ; kg:
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Since b0 P 0, a < 0 ) /0 ðaÞ 6 0, from the above inequality, we get b0 ðx; xÞ/0 ½fi ðxÞ fi ðxÞ 6 0:
ðMWDÞ max f ðyÞ ¼ ðf1 ðyÞ; f2 ðyÞ; . . . ; fk ðyÞÞ s:t:
Since b1 = 0 and a ¼ 0 ) /1 ðaÞ = 0, by the feasibility of x and (2), we get b1 ðx; xÞ/1 ½ lT gðxÞ lT gðxÞ 5 0:
f 0 ðx; gðx; xÞÞ þ lT g0 ðx; gðx; xÞÞ < 0; again a contradiction to (1). Assume that x is not a weak Pareto efficient solution of (P). Then there is a feasible solution x of (P) such that fi ðxÞ < fi ðxÞ for any i 2 f1; 2; . . . ; kg: Since b0 P 0 and a < 0 ) /0 ðaÞ < 0, from the above inequality, we get b0 ðx; xÞ/0 ½fi ðxÞ fi ðxÞ 6 0: Since b1 = 0 and a ¼ 0 ) /1 ðaÞ = 0, from feasibility of x and (2), we get b1 ðx; xÞ/1 ½ lT gðxÞ lT gðxÞ 5 0: By condition (c) and the above two inequalities, we get f 0 ðx; gðx; xÞÞ < 0 and lT g0 ðx; gðx; xÞÞ 6 0: By these two inequalities, we get f 0 ðx; gðx; xÞÞ þ lT g0 ðx; gðx; xÞÞ < 0; (1).
This
ð6Þ ð7Þ
nT e ¼ 1;
ð8Þ
n2
By the generalized d-univex condition in (b) and the above two inequalities, we get (4) and (5). From (4) and (5), we get
which contradicts proof. h
ðnT f 0 þ lT g0 Þðy; gðx; yÞÞ = 0 for all x 2 D; lj gj ðyÞ = 0; j ¼ 1; . . . ; m;
completes
the
Rkþ ;
l2
Rmþ ;
where e ¼ ð1; 1; . . . ; 1Þ 2 Rk . Denote by W ¼ ðy; n; lÞ 2 X Rk Rm : ðnT f 0 þ lT g0 Þðy; gðx; yÞÞ = 0; lj gj ðyÞ = 0; j ¼ 1; . . . ; m; n 2 Rkþ ; nT e ¼ 1; l 2 Rmþ denote the set of all the feasible solutions of (MWD). We denote by prX W the projection of set W on X . Theorem 2 (Weak duality). Let x and (y; n; l) be a feasible solution for (P) and (MWD), respectively. Moreover, suppose that one of the following conditions holds: (a) f is strong pseudo-d-univex at y on D [ prX W with respect to some b0 , /0 and g with n > 0, b0 > 0, a 6 0 ) /0 ðaÞ 6 0, and lT g is quasi dunivex at y on D [ prX W with respect to b1 , /1 and g with a 5 0 ) /1 ðaÞ 5 0; (b) f is weak strictly pseudo-d-univex at y on D [ prX W with respect to some b0 , /0 and g with b0 P 0, a 6 0 ) /0 ðaÞ 6 0, and lT g is quasi dunivex at y on D [ prX W with respect to b1 , /1 and g with a 5 0 ) /1 ðaÞ 5 0; (c) f is weak strictly pseudo-d-univex at y on D [ prX W with respect to some b0 , /0 and g with b0 P 0, a 6 0 ) /0 ðaÞ 6 0, and lT g is quasi dunivex at y on D [ prX W with respect to b1 , /1 and g with a 5 0 ) /1 ðaÞ 5 0. Then
4. Mond–Weir duality In relation to (P), we consider the following dual problem which is in the form of Mond–Weir [17]:
f ðxÞif ðyÞ:
Proof. We proceed by contradiction. Assume that f ðxÞ 6 f ðyÞ:
S.K. Mishra et al. / European Journal of Operational Research 160 (2005) 218–226
Since b0 > 0, a 6 0 ) /0 ðaÞ 6 0, from the above inequality, we get b0 ðx; yÞ/0 ½f ðxÞ f ðyÞ 6 0:
ð9Þ
Since x is feasible for (P) and (y; n; l) is feasible for (MWD), it follows that m m X X lj gj ðxÞ lj gj ðyÞ 5 0: j¼1
j¼1
Since b1 = 0, a 5 0 ) /1 ðaÞ 5 0, from the above inequality, we get " # m m X X b1 ðx; yÞ/1 lj gj ðxÞ lj gj ðyÞ 5 0: ð10Þ j¼1
j¼1
By the generalized d-univex condition in (a), (9) and (10) imply ð11Þ
and lj gj0 ðy; gðx; yÞÞ 5 0:
ð12Þ
j¼1
Since n > 0, from (11) and (12), we get k X
ni fi0 ðy; gðx; yÞÞ þ
i¼1
m X
lj gj0 ðy; gðx; yÞÞ < 0;
ð13Þ
l¼1
which contradicts (6). For the proof of part (b), again assume that f ðxÞ 6 f ðyÞ: Since b0 P 0, a 6 0 ) /0 ðaÞ 6 0, from the above inequality, we get b0 ðx; yÞ/0 ½f ðxÞ f ðyÞ 6 0:
ð14Þ
Since x is feasible for (P) and (y; n; l) is feasible for (MWD), it follows that m m X X lj gj ðxÞ lj gj ðyÞ 5 0: j¼1
f 0 ðy; gðx; yÞÞ < 0; and m X
lj gj0 ðy; gðx; yÞÞ 5 0:
j¼1
Since n = 0, the above two inequalities imply (13), again a contradiction to (6). For the proof of part (c), proceeding as in part (b), we get (14) and (15). By the generalized univexity condition in part (c), (14) and (15) imply f 0 ðy; gðx; yÞÞ < 0;
ð16Þ
and m X lj gj0 ðy; gðx; yÞÞ 6 0:
ð17Þ
j¼1
f 0 ðy; gðx; yÞÞ 6 0; m X
223
Since n = 0, (16) and (17) imply (13), again a contradiction to (6). This completes the proof. h Theorem 3 (Strong duality). Let x be a locally weak Pareto efficient solution or a weak Pareto efficient solution for (P) at which the generalized Slater’s constraint qualification is satisfied. Let (f ; g) be directionally differentiable at x with f 0 ðx; gðx; xÞÞ, and g0 ðx; gðx; xÞÞ preinvex functions on X and gj be continuous for j 2 b J ðxÞ. Then there exists l 2 Rmþ such that (x; 1; l) is feasible for (MWD). If the weak duality between (P) and (MWD) in Theorem 2 holds, then (x; 1; l) is a locally weak Pareto efficient solution for (MWD). Proof. Since x satisfies all the conditions of Lemma 4, there exists l 2 Rmþ such that conditions (1)–(3) hold. By (1)–(3), we have that (x; 1; l) is feasible for (MWD). By the weak duality, it follows that (x; 1; l) is a locally weak Pareto efficient solution for (MWD). h
j¼1
Since b1 = 0, a 5 0 ) /1 ðaÞ 5 0, from the above inequality, we get " # m m X X b1 ðx; yÞ/1 lj gj ðxÞ lj gj ðyÞ 5 0: ð15Þ j¼1
j¼1
By the generalized d-univex condition in (b), (14) and (15) imply
l) be a Theorem 4 (Converse duality). Let (y ; n; weak Pareto efficient solution for (MWD). If the hypothesis of Theorem 2 holds at y on D [ prX W , then y is a weak Pareto efficient solution for (P). Proof. We proceed by contradiction. Assume that y is not a weak Pareto efficient solution for (P), that is, there exists ~x 2 D such that
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f ð~xÞ < f ðy Þ: We know from condition (a) of Theorem 2 that b0 > 0, and a < 0 ) /0 ðaÞ < 0, from this and the above inequality, we get
Since ni = 0, the above two inequalities imply (20), again a contradiction to (6). This completes the proof. h
b0 ð~x; yÞ/0 ½f ð~xÞ f ðy Þ < 0:
5. General Mond–Weir duality
By the generalized univexity condition (a) in Theorem 2, we get
We shall continue our discussion on duality for (P) in the present section by considering a general Mond–Weir type dual problem and proving weak and strong duality theorems under an assumption of the generalized d-univexity introduced in Section 2. We consider the following general Mond–Weir type dual to (P):
k X
ni fi0 ðy ; gð~x; y ÞÞ < 0:
ð18Þ
i¼1
l) for From the feasibility of ~x for (P), (y ; n; (MWD), b1 = 0 and a 5 0 ) /1 ðaÞ 5 0, we have " # m m X X b1 ð~x; y Þ/1 lj gj ð~xÞ lj gj ðy Þ 5 0: j¼1
j¼1
The above inequality in light of the generalized dunivexity condition (a) in Theorem 2 yields m X ð19Þ lj gj0 ðy ; gð~x; y ÞÞ 5 0: j¼1
ni fi0 ðy ; gð~x; y ÞÞ þ
i¼1
m X
lj gj0 ðy ; gð~x; y ÞÞ < 0:
ð20Þ
j¼1
fi0 ðy ; gð~x; y ÞÞ
0, and (f þ lJ0 gJ0 ; lJt gJt ) is strong pseudoquasi d-type-I univex at y on D [ prX W with respect to some b0 ; b1 ; /0 ; /1 and g with b0 > 0, n > 0, a 6 0 ) /0 ðaÞ 6 0, and a 5 0 ) /1 ðaÞ 5 0 for any t, 1 5 t 5 r. (b) (f þ lJ0 gJ0 ; lJt gJt ) is weak strictly pseudoquasi d-type-I univex at y on D [ prX W with respect
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to some b0 , b1 , /0 , /1 and g with b0 P 0, a 6 0 ) /0 ðaÞ 6 0, and a 5 0 ) /1 ðaÞ 5 0 for any t, 1 5 t 5 r. (c) (f þ lJ0 gJ0 ; lJt gJt ) is weak strictly pseudod-type-I univex at y on D [ prX W with respect to some b0 ; b1 ; /0 , /1 and g with b0 P 0, a 6 0 ) /0 ðaÞ 6 0, and a 5 0 ) /1 ðaÞ 5 0 for any t, 1 5 t 5 r. Then the following cannot hold: f ðxÞ 6 /ðy; n; lÞ:
Similarly, by condition (b), we have 0
ðf þ lJ0 gJ0 0 eÞðy; gðx; yÞÞ < 0 and lJt gJ0 t ðy; gðx; yÞÞ 5 0 81 5 t 5 r: Since n = 0, the above two inequalities yield ! r X T 0 0 n f þ lJt gJt ðy; gðx; yÞÞ < 0: t¼0
Proof. We proceed by contradiction. Suppose that f ðxÞ 6 /ðy; n; lÞ:
ð24Þ
The above inequality leads to (28), which contradicts (21). By condition (c), we get ðf 0 þ lJ0 gJ0 0 eÞðy; gðx; yÞÞ < 0
Since x is feasible for (P) and l = 0, (24) implies that
and
f ðxÞ þ lTJ0 gJ0 ðxÞe 6 f ðyÞ þ lTJ0 gJ0 ðyÞe:
lJt gJ0 t ðy; gðx; yÞÞ < 0
Since b0 > 0, a 6 0 ) /0 ðaÞ 6 0, from the above inequality, we get h i b0 ðx;yÞ/0 f ðxÞ þ lTJ0 gJ0 ðxÞe f ðyÞ þ lTJ0 gJ0 ðyÞe 6 0:
Since n = 0, the above two inequalities yield ! r X T 0 0 lJt gJt ðy; gðx; yÞÞ < 0: n f þ
ð25Þ From the feasibility of x for (P) and (22), we have lTJt gJt ðxÞ lTJt gJt ðyÞ 5 0 for all 1 5 t 5 r: Since b1 = 0, a 5 0 ) /1 ðaÞ 5 0, from the above inequality, we get
b1 ðx;yÞ/1 lTJt gJt ðxÞ lTJt gJt ðyÞ 5 0 for all 1 5 t 5 r: ð26Þ By condition (a), from (25) and (26), we have ðf 0 þ lJ0 gJ0 0 eÞðy; gðx; yÞÞ 6 0 and lJt gJ0 t ðy; gðx; yÞÞ 5 0
225
81 5 t 5 r:
Since n > 0, the above two inequalities yield ! r X T 0 0 n f þ lJt gJt ðy; gðx; yÞÞ < 0: ð27Þ
81 5 t 5 r:
t¼0
The above inequality leads to (28), which contradicts (21). This completes the proof. h Theorem 6 (Strong duality). Let x be a locally weak Pareto efficient solution or a weak Pareto efficient solution for (P) at which the generalized Slater’s constraint qualification is satisfied. Let f and g be directionally differentiable at x with f 0 ðx; gðx; xÞÞ and g0 ðx; gðx; xÞÞ preinvex functions on X and gj be continuous for j 2 b J ðxÞ. Then, there exists l 2 Rmþ such that (x; 1; l) is feasible for (GMWD). Moreover, if the weak duality between (P) and (GMWD) in Theorem 5 holds, then (x; 1; l) is a locally weak Pareto efficient solution for (GMWD). Proof. The proof of this theorem is similar to the proof of Theorem 3. h
t¼0
Since J0 ; . . . ; Jr are partitions of M, (27) is equivalent to ðnT f 0 þ lT g0 Þðy; gðx; yÞÞ < 0; which contradicts the dual constraint (21).
ð28Þ
Acknowledgements The research was supported by the University Grants Commission of India, the National
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Natural Science Foundation of China and the Research Grants Council of Hong Kong.
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