SUFFICIENCY AND DUALITY FOR INTERVAL-VALUED OPTIMIZATION PROBLEMS WITH VANISHING CONSTRAINTS USING WEAK CONSTRAINT QUALIFICATIONS

In this paper, we are concerned with one of the difficult class of optimization problems called the interval-valued optimization problem with vanishing constraints. Sufficient optimality conditions for a LU optimal solution are derived under generalized convexity assumptions. Moreover, appropriate duality results are discussed for a Mond-Weir type dual problem. In addition, numerical examples are given to support the sufficient optimality conditions and weak duality theorem.


Introduction
Due to the mathematical challenges and important roles in various fields, mathematical programs with vanishing constraints have attracted many mathematicians in the past decade. Mathematical programming problem with vanishing constraints is a constrained optimization problem and it is closely related to the Mathematical programs with equilibrium constraints, see for example [9,10,14]. This problem was first studied by Achtziger and Kanzow in [2] and this serves as a model for many problems from topology and structural optimization (see [2,5]). For Mathematical programming problems with vanishing constraints, it is well known that the usual nonlinear programming constraint qualifications such as Slater constraint qualification, Mangasarian-Fromovitz constraint qualification, Cottle constraint qualification and linear independence constraint qualification do not hold (see [12]), while Mishra et al. [12] proved that the standard generalized Guignard constraint qualification holds in many situations, and some sufficient conditions are presented in [12].
The Guignard constraint qualification (GCQ) was introduced by Guignard [8] and is one of the weakest among the most prominent constraint qualifications such as the Slater constraint qualification [19], Abadie constraint qualification [1], Mangasarian-Fromovitz constraint qualification [11], Cottle constraint qualification [7] and linear independence constraint qualification etc. For more information and inter-relation between these constraint qualifications one can see the survey papers [15,22].
In recent years, a number of approaches have been developed to deal with interval-valued optimization problems. In [24,25], Wu derived Karush-Kuhn-Tucker type optimality conditions for a optimization problem with an interval-valued objective function. Further, the Karush-Kuhn-Tucker type necessary optimality conditions for a optimization problem in which objective and constraints functions are assumed to be interval valued were investigated by Singh et al. [17]. However, optimality conditions for an interval-valued multiobjective programming with generalized differentiable functions (viz. gH-differentiable functions) are discussed in [18]. Bhurjee and Panda [6] provided an overview of an interval-valued optimization problem by developing a methodology to study the efficient solution for an interval-valued optimization problem. For more details related to interval-valued optimization problems, we refer to the papers (see, for example [3,13,16,20,23,26]).
To the author's knowledge, there are no results for an interval-valued mathematical programming problem with vanishing constraints in the literature. Therefore, this paper focuses on an interval-valued mathematical programming problem with vanishing constraints to explore the sufficient optimality conditions and Mond-Weir type duality results.
The rest of the article is organized as follows: Some background material and preliminary definitions are provided in Section 2. The sufficient optimality conditions for a LU optimal solution for considered problem under generalized convexity assumptions are given in Section 3. In Section 4, weak, strong and strict converse duality theorems are discussed for a Mond-Weir type dual model. Finally, Section 5 is devoted to the conclusion.

Preliminaries
For a nonempty subset Q of R n , we use the notations clQ and clcoQ to denote the closure of Q and closure of the convex hull of Q, respectively. Let Θ be the set of all closed and bounded intervals in R. Let where k is a real number.
Let x * ∈ F be any feasible solution of the (IVVC). The following index sets will be used in the sequel.
Furthermore, the index set Λ + can be divided into the following subsets Similarly, the index set Λ 0 can be partitioned in the following way Also, for x * ∈ F, we define the sets Q k , Q k , k = L, U and Q as follows: The linearizing cone Q k , k = L, U at x * ∈ F is given by and the symbol T denotes the transpose of a matrix. The linearizing cone to Q at x * ∈ Q, given by.
Definition 2.2. The tangent cone to Q at x * ∈ clQ is defined by The modified Guignard constraint qualification was introduced by Mishra et al. ( [12], Definition 6.14) for a mathematical programming problem with vanishing constraints. From this perspective, we define the modified Guignard constraint qualification (IVVC-GCQ) for an interval-valued optimization problem (IVVC) as follows.
Mishra et al. [12] proved the Karush-Kuhn-Tucker type necessary optimality conditions for a multiob- Theorem 2.1. Let x * ∈ F be a LU optimal solution of (IVVC) such that (IVVC-GCQ) holds at x * . Then We define the following index sets which will be useful to prove the sufficient optimality conditions and duality results.
We now turn our attention to define some well-known concepts of convexity and generalized convexity for a real valued differentiable function (see, for example, [4]).

Definition 2.4.
Let Ω : X ⊆ R n → R be a continuously differentiable function. Then, Ω is said to be a (strictly) convex at (x = x * ∈ X) x * ∈ X if for any x ∈ X, we have Definition 2.5.
Let Ω : X ⊆ R n → R be a continuously differentiable function. Then, Ω is said to be a quasiconvex at x * ∈ X if for any x ∈ X, we have Definition 2.6. Let Ω : X ⊆ R n → R be a continuously differentiable function. Then, Ω is said to be a (strictly) pseudoconvex at x * ∈ X if for any x ∈ X, we have equivalently

Sufficient optimality conditions
In this section, we establish sufficient optimality conditions for the problem (IVVC) using the concept of generalized convexity.

5)
Further, assume that λ L Ψ L (.) + λ U Ψ U (.) is pseudoconvex atx on F and that Thenx is a LU optimal solution of the problem (IVVC).
Proof. Suppose contrary to the result thatx is not a LU optimal solution to the problem (IVVC), then by .
Since λ L > 0, λ U > 0, therefore the above inequalities yield which by pseudoconvexity of λ L Ψ L (.) + λ U Ψ U (.) atx on F, we obtain For x 0 ∈ F, µ i ∈ R + , i = 1, 2, ..., p, we have µ i ϕ i (x 0 ) ≤ 0, i = 1, 2, ..., p, which in view of (3.2) implies that which by quasiconvexity of By similar arguments, we have which by the definition of index sets one has On adding (3.7), (3.8) and (3.9), we get which contradicts (3.1). This completes the proof of this theorem. Now, we verify the sufficient optimality conditions by the following example.
Since all the assumptions of Theorem 3.1 are satisfied, thenx = 0 is a LU optimal solution of the problem (IVVC-1).

Mond-Weir type duality
We present the following Mond-Weir type dual for (IVVC).
Since λ L > 0, λ U > 0, therefore the above inequalities yield which by pseudoconvexity of λ L Ψ L (.) + λ U Ψ U (.) at y on F ∪ prW 1 , we obtain which by quasiconvexity of By similar arguments, we have which by the definition of index sets one has On adding (4.8), (4.9) and (4.10), we get which contradicts (4.1). This completes the proof of this theorem. Now, we verify the weak duality theorem by the following example.

Conclusion
In this paper, we have derived sufficient optimality conditions for an interesting class of interval-valued optimization problems with vanishing constraints under generalized convexity assumptions. Furthermore, weak, strong and strict converse duality results for a Mond-Weir type dual model have been established. It would be interesting to see whether the results derived in this paper hold for a non-differentiable multiple interval-valued objective programming problems with vanishing constraints. We shall investigate it in our forthcoming papers.