On optimality conditions for multiobjective ... - Science Direct

J. Math. Anal. Appl. 334 (2007) 123–131 www.elsevier.com/locate/jmaa

On optimality conditions for multiobjective optimization problems in topological vector space Cristinca Fulga a,∗ , Vasile Preda b a Academy of Economics Studies, Faculty of Cybernetics, Statistics and Economic Informatics,

Department of Mathematics, Piata Romana 6, Bucharest, Romania b University of Bucharest, Faculty of Mathematics and Computer Sciences, street Academiei 14,

Bucharest 010014, Romania Received 7 May 2006 Available online 29 December 2006 Submitted by I. Lasiecka

Abstract In this paper, we are concerned with a differentiable multiobjective programming problem in topological vector spaces. An alternative theorem for generalized K subconvexlike mappings is given. This permits the establishment of optimality conditions in this context: several generalized Fritz John conditions, in line to those in Hu and Ling [Y. Hu, C. Ling, The generalized optimality conditions of multiobjective programming problem in topological vector space, J. Math. Anal. Appl. 290 (2004) 363–372] are obtained and, in the presence of the generalized Slater’s constraint qualification, the Karush–Kuhn–Tucker necessary optimality conditions. © 2006 Elsevier Inc. All rights reserved. Keywords: Multiobjective programming; Optimality conditions; Topological vector spaces

1. Introduction Optimality conditions for optimization problems have been studied extensively in the literature. In the last years, attempts have been made to weaken the convexity hypotheses and thus to explore the extent of optimality conditions applicability in mathematical programming. Hanson [3] introduced the concept of invexity as a generalization of convexity for scalar constrained op* Corresponding author.

E-mail address: [email protected] (C. Fulga). 0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.12.047

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C. Fulga, V. Preda / J. Math. Anal. Appl. 334 (2007) 123–131

timization problems. Later, a further generalization, called preinvex, was considered by Hanson and Mond [4], and first optimality conditions and duality for multiobjective problems were given by Weir and Mond [11]. The multiobjective optimization problems in topological vector spaces were studied in Hu and Ling [5] for the K subconvexlike case. Other new developments in multiobjective programming under generalized convexity assumptions were made, see for example Suneja and Srivastava [10], Aghezzaf and Hachimi [1], Mishra et al. [8], Bector and Singh [2]. This paper deals with the multiobjective programming problem in topological vector spaces. In Section 2 we present extensions of the concepts of subconvex and subconvexlike functions and we establish the relations between these classes. In Section 3 we establish a theorem of alternative in the previously presented context. In Sections 4 and 5, several generalized Fritz John necessary conditions for that a feasible point be a weakly K-efficient solution of the problem are proved; then, by using the generalized Slater’s constraint qualification, some generalized Karush–Kuhn– Tucker necessary conditions are obtained. In the former, the main results concern the case when the mappings involved are generalized K subconvexlike and in the latter the results concern mappings with generalized K subconvexlike derivatives. We consider three topological vector spaces Ω, Γ and Γ1 . Let K ⊂ Γ and K1 ⊂ Γ1 be two cones; throughout this paper, all cones will be convex with the apex at the origin in the vector space. The orders in Γ and Γ1 are determined by K and K1 , respectively. The cone K it is assumed to have int K = ∅. In this paper we will consider the multiobjective programming problem in topological vector space: min f (x), (1.1) s.t. −g(x) ∈ K1 , where f : Ω → Γ and g : Ω → Γ1 are objective mapping and constraint mapping. By X we will denote the set of all feasible solutions of problem (1.1) which is assumed to be nonempty: X = x ∈ Ω −g(x) ∈ K1 . (1.2) The minimization in problem (1.1) means obtaining K-efficient solutions, respectively, weakly K-efficient solutions in the sense of the definition given below. Definition 1.1. A point x ∈ X is called a K-efficient solution of the problem (1.1) if there exists no x ∈ X such that f (x) − f (x) ∈ int K \ {0}. Definition 1.2. A point x ∈ X is called a weakly K-efficient solution of the problem (1.1) if there exists no x ∈ X such that f (x) − f (x) ∈ int K. 2. Preliminaries Let Π be a topological vector space and C ⊂ Π , a convex cone with int C = ∅. Let Π ∗ denote the topological dual of Π . The conjugate cone C ∗ of C is defined by C ∗ = c∗ ∈ Π ∗ c∗ , c 0, ∀c ∈ C (2.1) where c∗ , c is the value of linear functional c∗ at c.


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Remark 2.1. Using the definition of the conjugate cone C ∗ it is easy to prove that if c∗ ∈ C ∗ \ {0} and c ∈ int C, then c∗ , c > 0. Let S ⊂ Ω be a nonempty set, and h : S → Π be a mapping; let C ⊂ Π be a convex cone with int C = ∅. Definition 2.1. (See [5].) Let S be a convex set. The mapping h : S → Π is called C subconvex on S if there exists v ∈ int C such that, for any λ ∈ (0, 1), ε > 0 and for any x 1 , x 2 ∈ S, εv + λh x 1 + (1 − λ)h x 2 − h λx 1 + (1 − λ)x 2 ∈ C. Definition 2.2. Let S be a convex set. The mapping h : S → Π is called generalized C subconvex on S if there exists v ∈ int C, for any λ ∈ (0, 1), ε > 0 and for any x 1 , x 2 ∈ S, there exists ρ > 0 such that εv + λh x 1 + (1 − λ)h x 2 − ρh λx 1 + (1 − λ)x 2 ∈ C. Definition 2.3. (See [6].) Let S ⊂ Ω be a nonempty set. The mapping h : S → Π is called C subconvexlike on S if there exists v ∈ int C, for any λ ∈ (0, 1), ε > 0 and for any x 1 , x 2 ∈ S, there exists u ∈ S such that εv + λh x 1 + (1 − λ)h x 2 − h(u) ∈ C. Definition 2.4. Let S ⊂ Ω be a nonempty set. The mapping h : S → Π is called generalized C subconvexlike on S if there exists v ∈ int C, for any λ ∈ (0, 1) and ε > 0 and for any x 1 , x 2 ∈ S, there exist ρ > 0 and u ∈ S such that εv + λh x 1 + (1 − λ)h x 2 − ρh(u) ∈ C. The concepts that we will define in the sequel, demand that we recall the definition of an η-invex set. Definition 2.5. (See [9].) Let S ⊂ Ω be a nonempty set. We call a set S ⊂ Ω invex with respect to a given η : Ω × Ω → Ω (or η-invex) if x 1 , x 2 ∈ S, λ ∈ [0, 1] ⇒ x 2 + λη x 1 , x 2 ∈ S. The definition essentially says that there is a path starting from x 2 which is contained in S. It is not required that x 1 should be one of the end points of the path; if we demand that x 1 should be an end point of the path for every pair x 1 , x 2 ∈ S, then η(x 1 , x 2 ) = x 1 − x 2 , reducing to convexity. Next, we consider the following classes of functions defined on η-invex sets. Definition 2.6. Let S be a nonempty η-invex set, S ⊂ Ω and h : S → Π be a mapping; let C ⊂ Π be a convex cone with int C = ∅. The mapping h : S → Π is called C − η subconvex on S if there exists v ∈ int C, for any λ ∈ (0, 1) and ε > 0 and for any x 1 , x 2 ∈ S, εv + λh x 1 + (1 − λ)h x 2 − h x 2 + λη x 1 , x 2 ∈ C.

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Definition 2.7. Let S be a nonempty η-invex set, S ⊂ Ω and h : S → Π be a mapping; let C ⊂ Π be a convex cone with int C = ∅. The mapping h : S → Π is called generalized C − η subconvex on S if there exists v ∈ int C, for any λ ∈ (0, 1) and ε > 0 and for any x 1 , x 2 ∈ S, there exists ρ > 0 such that εv + λh x 1 + (1 − λ)h x 2 − ρh x 2 + λη x 1 , x 2 ∈ C. Remark 2.2. Let S ⊂ Ω be a nonempty set. We suppose that S is an invex set with respect to a given mapping η : Ω × Ω → Ω for the implications that requires this condition. For a mapping h : S → Π , we have C subconvex ⇓ generalized C subconvex

⇒ C − η subconvex ⇓ generalized ⇒ C − η subconvex

⇒ ⇒

C subconvexlike ⇓ generalized C subconvexlike

The example bellow illustrates that a generalized C subconvexlike mapping may not be C subconvexlike. Example 2.1. Let

2 2 S = (x, y) ∈ R 2 y = 2 − 2x, x 1 ∪ (x, y) ∈ R 2 x = 2 − 2y, y 1 . 3 3 Let h : S → R 2 be defined by h(x, y) = (x, y), ∀(x, y) ∈ S, and let the convex cone be C = [0; ∞) × [0; ∞). Then, h is generalized C subconvexlike on S but it is not C subconvexlike since h(S) + int C is not a convex set (see Theorem 2.1 in [7]). 3. A theorem of the alternative We first prove the following auxiliary result. Lemma 3.1. Let h : S → Π be a mapping generalized C subconvexlike on S. Then, cone(h(S)) + int C is a convex set. Proof. Let y 1 , y 2 ∈ cone(h(S)) + int C, then there exist α1 , α2 > 0, x 1 , x 2 ∈ S and c1 , c2 ∈ int C such that y i = αi h(x i ) + ci , i = 1, 2. For λ > 0 we have (3.1) λy 1 + (1 − λ)y 2 = λα1 + (1 − λ)α2 βh x 1 + (1 − β)h x 2 + c , λα1 ∈ (0, 1) and c = λc1 + (1 − λ)c2 . Since int C is a convex cone, we where β = λα1 +(1−λ)α 2 have c ∈ int C. It follows that there exists a neighborhood V of the origin O of Π such that c + V ⊂ int C. Since h is generalized C subconvexlike on S, for any ε > 0 there exist v ∈ int C, u ∈ S and ρ > 0 such that (3.2) εv + λh x 1 + (1 − λ)h x 2 − ρh(u) ∈ C.

But the neighborhood V is absorbent so, we may choose ε which is sufficiently small such that −ε(λα1 + (1 − λ)α2 )v ∈ V . For such ε we have c − ε λα1 + (1 − λ)α2 v ∈ c + V ⊂ int C. (3.3)


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From (3.1) and (3.2) and with the notation α = λα1 + (1 − λ)α2 , we have that there is c ∈ C such that λy 1 + (1 − λ)y 2 = α c − εv + ρh(u) + c . (3.4) Rearranging (3.4) and combining with (3.3) we get c − εαv + αc + ραh(u) ∈ c + V + C + cone h(S) ⊂ int C + cone h(S) . Therefore we obtained that λy 1 + (1 − λ)y 2 ∈ cone(h(S)) + int C so, cone(h(S)) + int C is a convex set. 2 Now, we are ready to prove a theorem of alternative in the context of generalized C subconvexlike mappings. Theorem 3.1. Let Ω be a topological vector space and S ⊂ Ω a nonempty set. Let Π be a topological vector space and C ⊂ Π a convex cone with int C = ∅. If h : S → Π is a generalized C subconvexlike mapping on S, then either (i) or (ii) holds: (i) −h(S) ∩ int C = ∅, (ii) ∃c∗ ∈ C ∗ − {0} such that c∗ , h(x) 0, ∀x ∈ S. The two alternatives (i) and (ii) exclude each other. Proof. First, we suppose that (i) holds. Hence, there is x ∈ S such that −h(x) ¯ ∈ int C. From Remark 2.1 we have c∗ , h(x)

¯ < 0, ∀c∗ ∈ C ∗ \ {0}, so, (ii) does not hold. Now, we suppose that (i) does not hold, i.e. −h(S) ∩ int C = ∅. In this case, we have −(h(S) + int C) ∩ int C = ∅ and therefore −(cone(h(S)) + int C) ∩ int C = ∅. Hence, since by Lemma 3.1 cone(h(S)) + int C is convex, by the separation theorem there exists c∗ ∈ Π ∗ \ {0} such that ∗ (3.5) c , y 0 c∗ , c , ∀c ∈ int C, ∀y ∈ − cone h(S) + int C . First, combining (2.1) with (3.5) we get c∗ ∈ C ∗ \ {0}. For any y ∈ −(cone(h(S)) + int C) there exist ν > 0, x¯ ∈ S and q¯ ∈ int C such that y = −(νh(x) ¯ + q). ¯ From (3.5) we have ∗ − c , h(x) ¯ μ c∗ , q¯ , (3.6) where μ = ν1 > 0. We consider {μn }, μn > 0, ∀n ∈ N , μn 0. From (3.6) we have − c∗ , h(x)

¯ μn c∗ , q ; ¯ by taking the limit, we obtain c∗ , h(x)

¯ 0, ∀x¯ ∈ S, and (ii) holds. 2 4. Optimality conditions under generalized subconvexlike hypothesis for the mapping pair (f, g) Before the main results, we recall some definitions and make some remarks. First, we recall the Gateaux derivative of a mapping.

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Definition 4.1. Let Ω and Π be two topological vector spaces. The mapping h : Ω → Π is called Gateaux differentiable at x¯ if for any x ∈ Ω, there exists the limit h(x¯ + αx) − h(x) ¯ . α→0 α

hx¯ (x) = lim

The mapping x → hx¯ (x), x ∈ Ω, is called the Gateaux derivative of h at x. ¯ Remark 4.1. For x = 0 ∈ Ω, the Gateaux derivative at x¯ is null. Definition 4.2. Let Ω, Γ and Γ1 be three topological vector spaces and S ⊂ Ω a nonempty set. Let f : S → Γ and f1 : S → Γ1 be two mappings and F (x) = (f (x), f1 (x)), ∀x ∈ S. Let K ⊂ Γ and K1 ⊂ Γ1 be two convex cones with nonempty interiors. The mapping pair (f, f1 ) is called (K × K1 )-subconvexlike on S if F is (K × K1 )-subconvexlike on S. Remark 4.2. In the same context, the mapping pair (f, f1 ) is called generalized (K × K1 )subconvexlike on S if F is generalized (K × K1 )-subconvexlike on S. Now, we state the necessary conditions for the feasible point x¯ to be weakly K-efficient solution of the problem (1.1) where the functions involved are generalized (K × K1 )-subconvexlike. Theorem 4.1. Let f : Ω → Γ and g : Ω → Γ1 be Gâteaux differentiable at x ∈ X, where X is the set of feasible solutions defined by (1.2) and let (f, g) be generalized (K × K1 )-subconvexlike on Ω. If x is a weakly K-efficient solution of the problem (1.1), then there exist γ ∗ ∈ Γ ∗ and γ1∗ ∈ Γ1∗ such that ∗ ∗ γ , fx (x) + γ1 , gx (x) = 0, ∀x ∈ Ω, (4.1) ∗ γ1 , g(x) = 0, (4.2) ∗T ∗T T ∗ ∗ ∗ ∗ γ ∈K , γ1 ∈ K1 , γ , γ1 = 0. (4.3) Proof. Let h(x) = (f (x) − f (x), g(x)), x ∈ Ω. Since x is a weakly K-efficient solution there exists no x ∈ Ω such that −g(x) ∈ K1 and f (x) − f (x) ∈ int K, i.e. there exists no x ∈ Ω such that h(x) ∈ − int(K × K1 ). From Remark 4.2 we have that h is generalized (K × K1 )T T subconvexlike on Ω. According to Theorem 3.1, there exists (γ ∗ , γ1∗ )T ∈ (K ∗ × K1∗ ) \ {(0, 0)} T T such that (γ ∗ , γ1∗ )T , h(x) 0, ∀x ∈ Ω. Thus, we have ∗ (4.4) γ , f (x) − γ ∗ , f (x) + γ1∗ , g(x) 0, ∀x ∈ Ω. Taking x = x in (4.4), ∗ γ1 , g(x) 0.

(4.5)

Since γ1∗ ∈ K1∗ and −g(x) ∈ K1 , from (2.1) we have γ1∗ , −g(x) 0; combining with (4.5) we obtain (4.2). From (4.4) and (4.2) we get ∗ γ , f (x) − f (x) + γ1∗ , g(x) − g(x) 0, ∀x ∈ Ω. For α > 0 we have ∗ f (x + αx) − f (x) ∗ g(x + αx) − g(x) + γ1 , 0, γ , α α

∀x ∈ Ω.


Letting α → 0, we obtain ∗ ∗ γ , fx (x) + γ1 , gx (x) 0,

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∀x ∈ Ω.

On the other hand, using the properties of the Gâteaux derivative, we get γ ∗ , fx (x) + γ1∗ , gx (x) 0, ∀x ∈ Ω, and therefore (4.1). 2 Corollary 4.2. If the hypotheses of Theorem 4.1 are satisfied, let (f, g) be generalized (K × K1 )subconvex on Ω. If x is a weakly K-efficient solution of the problem (1.1), then there exist γ ∗ ∈ Γ ∗ and γ1∗ ∈ Γ1∗ such that (4.1)–(4.3) hold. The proof of the corollary is immediate if we combine Theorem 4.1 and Remark 2.2. Remark 4.3. Because of Remark 2.2 we find that Theorem 3.1 and Corollary 3.2 in [5] are consequences of Theorem 4.1. Definition 4.3. Let g : Ω → Γ1 be the constraint mapping in problem (1.1), K1 ⊂ Γ1 be a convex cone with the vertex OΓ1 and with int K1 = ∅. Let X, the set of all feasible solutions of problem (1.1), be a nonempty set. We say that g satisfies the generalized Slater constraint qualification if there is x ∈ X such that −g(x) ∈ int K1 . Theorem 4.3. Let f : Ω → Γ and g : Ω → Γ1 be Gâteaux differentiable at x ∈ X, where X is the set of feasible solutions defined by (1.2) and let (f, g) be generalized (K × K1 )-subconvexlike on Ω. If g satisfies the generalized Slater constraint qualification and x is a weakly K-efficient solution of the problem (1.1), then there exist γ ∗ ∈ Γ ∗ and γ1∗ ∈ Γ1∗ such that ∗ ∗ γ , fx (x) + γ1 , gx (x) = 0, ∀x ∈ Ω, (4.6) ∗ (4.7) γ1 , g(x) = 0, γ ∗ ∈ K ∗ \ {0},

γ1∗ ∈ K1∗ .

(4.8)

Proof. Previously we have proved Theorem 4.1 therefore, we have only to show that γ ∗ = 0. Suppose the contrary, γ ∗ = 0. Hence, it follows from (4.4) that γ1∗ , g(x) 0, ∀x ∈ Ω. Since γ ∗ = 0, according to (4.3), γ1∗ = 0. Due to the fact that g satisfies the generalized Slater constraint qualification, we have that there exists x ∈ Ω such that γ1∗ , −g(x ) > 0 which is a contradiction. 2 Corollary 4.4. If the hypotheses of Theorem 4.3 are satisfied, let (f, g) be generalized (K × K1 )subconvex on Ω. If x is a weakly K-efficient solution of the problem (1.1), then there exist γ ∗ ∈ Γ ∗ and γ1∗ ∈ Γ1∗ such that (4.6)–(4.8) hold. 5. Optimality conditions under generalized subconvexlike hypothesis for the mapping pair (fx , gx ) The next results concern optimality conditions for the problem (1.1) under generalized subconvexlike hypothesis for the mapping pair of the derivatives. Theorem 5.1. Let f : Ω → Γ and g : Ω → Γ1 be Gâteaux differentiable at x ∈ X, where X is the set of feasible solutions defined by (1.2) and let (fx , gx ) be generalized (K × K1 )-subconvexlike

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on Ω. If x is a weakly K-efficient solution of the problem (1.1), then there exist γ ∗ ∈ Γ ∗ and γ1∗ ∈ Γ1∗ such that (4.1)–(4.3) hold. Proof. Let ψ(x) = (fx (x), gx (x) + g(x)), x ∈ Ω. Suppose that there is x ∈ Ω such that ψ(x) ∈ − int(K × K1 ), we have fx (x) ∈ − int K,

(5.1)

gx (x) + g(x) ∈ − int K1 .

(5.2)

From (5.2) we have that there exits β > 0 such that g(x + αx) − g(x) + g(x) ∈ − int K1 , ∀α ∈ (0, β). (5.3) α Since K1 is a convex cone, from (5.3) we get −g(x + αx) ∈ K1 , i.e. x + αx ∈ X. In a similar way, from (5.1) we obtain f (x) − f (x + αx) ∈ int K that contradicts the fact that x is a weakly K-efficient solution of the problem (1.1). So, ψ(x) ∈ / − int(K × K1 ), ∀x ∈ Ω, and, from TheT T T T orem 4.1, there exists (γ ∗ , γ1∗ )T ∈ (K ∗ × K1∗ ) \ {(0, 0)} such that (γ ∗ , γ1∗ )T , ψ(x) 0, ∀x ∈ Ω. Thus, we have ∗ ∗ γ , fx (x) − γ1 , gx (x) + γ1∗ , g(x) 0, ∀x ∈ Ω. (5.4) Taking x = 0 in (5.4) we get γ1∗ , g(x) 0. Since x ∈ X and by the definition of the conjugate cone, we have γ1∗ , −g(x) 0, so we obtained (4.2). From (5.4) and (4.2) we get γ ∗ , fx (x) − γ1∗ , gx (x) 0, ∀x ∈ Ω. Since the Gateaux differentiable at x¯ for the 0-vector is null, we infer that γ ∗ , fx (x) − γ1∗ , gx (x) 0, ∀x ∈ Ω. So, we get (4.1). 2 Corollary 5.2. If the hypotheses of Theorem 5.1 are satisfied, let (fx , gx ) be generalized (K × K1 )-subconvex on Ω. If x is a weakly K-efficient solution of the problem (1.1), then there exist γ ∗ ∈ Γ ∗ and γ1∗ ∈ Γ1∗ such that (4.1)–(4.3) hold. Theorem 5.3. Let f : Ω → Γ and g : Ω → Γ1 be Gâteaux differentiable at x ∈ X, where X is the set of feasible solutions defined by (1.2) and let (fx , gx ) be generalized (K × K1 )-subconvexlike on Ω. If x is a weakly K-efficient solution of the problem (1.1), g is K1 − η subconvex on Ω and satisfies the generalized Slater constraint qualification, then there exist γ ∗ ∈ Γ ∗ and γ1∗ ∈ Γ1∗ such that (4.6)–(4.8) hold. Proof. Due to Theorem 5.1, we have to prove only γ ∗ = 0. Suppose the contrary, γ ∗ = 0. Since γ ∗ = 0, according to (4.3), γ1∗ = 0. Hence, it follows from (4.4) that γ1∗ , gx (x) 0, ∀x ∈ Ω. But, −x ∈ Ω, so, we obtain ∗ γ1 , gx (x) = 0, ∀x ∈ Ω. (5.5) Since g satisfies the generalized Slater constraint qualification, there exists x ∈ Ω such that ∗ γ1 , g(x ) < 0. (5.6) By the definition of K1 − η subconvexity, there exists v ∈ int K1 , for any λ ∈ (0, 1) and ε > 0 and for any x 1 , x 2 ∈ Ω, ελv + λg(x ) + (1 − λ)g(x) − g x + λη(x , x) ∈ K1 .


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It follows that g(x + λη(x , x)) − g(x) ∈ K1 . λ Taking λ 0, we find εv + g(x ) − g(x) − gx (η(x , x)) ∈ K1 and therefore, ε γ1∗ , v + γ1∗ , g(x ) − γ1∗ , −g(x) − γ1∗ , gx η(x , x) 0. εv + g(x ) − g(x) −

Combining with (4.7) and (5.5), it follows that, if ε 0, we obtain γ1∗ , g(x ) 0 which contradicts (5.6). 2 Corollary 5.4. If the hypotheses of Theorem 5.3 are satisfied, let (fx , gx ) be generalized (K × K1 )-subconvex on Ω. If x is a weakly K-efficient solution of the problem (1.1), g is K1 − η subconvex on Ω and satisfies the generalized Slater constraint qualification, then there exist γ ∗ ∈ Γ ∗ and γ1∗ ∈ Γ1∗ such that (4.6)–(4.8) hold. References [1] B. Aghezzaf, M. Hachimi, Sufficient optimality conditions and duality in multiobjective optimization involving generalized convexity, Numer. Funct. Anal. Optim. 22 (7/8) (2001) 775–788. [2] C.R. Bector, C. Singh, B-vex functions, J. Optim. Theory Appl. 71 (1991) 237–253. [3] M. Hanson, On sufficiency of Kuhn–Tucker conditions, J. Math. Anal. Appl. 80 (1981) 545–550. [4] M. Hanson, B. Mond, Necessary and sufficient conditions in constrained optimization, Math. Program. 37 (1987) 51–58. [5] Y. Hu, C. Ling, The generalized optimality conditions of multiobjective programming problem in topological vector space, J. Math. Anal. Appl. 290 (2004) 363–372. [6] V. Jeyakumar, A generalization of a minimax theorem of Fan via a theorem of the alternative, J. Optim. Theory Appl. 48 (1986) 525–533. [7] Z. Li, A theorem of the alternative and its application to the optimization of set-valued maps, J. Optim. Theory Appl. 100 (1999) 365–375. [8] S.K. Mishra, S.Y. Wang, K.K. Lai, Nondifferentiable multiobjective programming under generalized d-univexity, European J. Oper. Res. 160 (2005) 218–226. [9] S.R. Mohan, S.K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl. 189 (1995) 901–905. [10] S.K. Suneja, M.K. Srivastava, Optimality and duality in nondifferentiable multiobjective optimization involving d-type I and related functions, J. Math. Anal. Appl. 206 (1997) 465–479. [11] T. Weir, B. Mond, Preinvex functions in multiple objective optimization, J. Math. Anal. Appl. 136 (1988) 29–38.