Duality theorems for nondifferentiable semi-infinite interval-valued optimization problems with vanishing constraints

In this paper, we study the duality theorems of a nondifferentiable semi-infinite interval-valued optimization problem with vanishing constraints (IOPVC). By constructing the Wolfe and Mond–Weir type dual models, we give the weak duality, strong duality, converse duality, restricted converse duality, and strict converse duality theorems between IOPVC and its corresponding dual models under the assumptions of generalized convexity.


Introduction
In recent years, the mathematical programming problems with vanishing constraints (MPVCs) have been studied extensively by many scholars. Achtziger and Kanzow [1] first proposed an optimization problem with vanishing constraints (MPVCs) and gave the strong stationary point theorem and VC-stationary point theorem of MPVCs under ACQ and improved ACQ assumptions. Under the inspiration of [1], Hoheisel and Kanzow [2] gave the M-stationary point theorem of MPVCs by using the MPVCs-GCQ. Guu et al. [3] studied the strong KKT type optimality conditions for nonsmooth multiobjective semiinfinite programming problems with vanishing constraints by the generalized S-stationary and M-stationary point conditions. Tung [4] studied the necessary and sufficient KKT type optimality conditions for continuously differentiable multiobjective semi-infinite MPVCs by using the ACQ and VC-ACQ in [1].
Accordingly, the study of dual problems related to MPVCs has also been used as a tool to solve optimization problems in various fields in the past decades, such as variational problems, fractional programming problems, semi-infinite programming problems, complex minimax problems, and so on. Tung [4] presented Wolfe and Mond-Weir type dual models for differentiable multiobjective semi-infinite programming with vanishing constraints and discussed the weak and strong duality theorems. Mishra and Singh [5] studied the continuously differentiable MPVCs and gave the weak, strong, converse, restricted converse, and strict converse duality theorems between MPVCs and the corresponding Wolfe and Mond-Weir type dual models under the assumptions of convexity and strict convexity. Hu and Wang et al. [6] proposed a new Wolfe and Mond-Weir type dual models related to continuously differentiable MPVCs and studied the weak, strong, converse, restricted converse, and strict converse duality theorems between them under the assumptions of convexity and generalized convexity.
With the development of mathematics, there are more and more researchers paying their attention to interval-valued optimization problems. Wu [7] studied the Wolfe type dual problem for continuously differentiable interval-valued optimization problems. Sun and Wang [8] gave the optimality conditions and duality for nondifferentiable intervalvalued optimization problems. Tung [9] studied the optimality conditions and duality for convex semi-infinite multiobjective interval-valued optimization problems. Ahmad et al. [10] studied continuously differentiable interval-valued variational problem and gave the sufficient optimality condition and Mond-Weir type duality of the original problem by using the invexity conditions. Kummari and Ahmad [11] discussed the optimality conditions and duality for nonsmooth interval-valued optimization problems with equality and inequality constraints via the L-invex-infine functions. Jayswal et al. [12,13] gave the optimality conditions and duality for nonsmooth interval-valued optimization problems with inequality constraints by using generalized convexity. Su and Dinh [14] studied the duality for interval-valued pseudoconvex optimization problem with equilibrium constraints by using the notion of contingent epiderivatives. Recently, Ahmad et al. [15] studied the optimality conditions and Mond-Weir type dual problems for differentiable interval-valued optimization problems with vanishing constraints.
Inspired by the literatures mentioned above, in this paper, we study the duality theorems for nondifferentiable semi-infinite interval-valued optimization problem with vanishing constraints(IOPVC) and explore the dual relationships between IOPVC and its corresponding Wolfe and Mond-Weir type dual models. The paper is organized as follows. In Sect. 2, we introduce some known concepts and formulas; In Sect. 3, we study the weak, strong, converse, restricted converse, and strict converse duality theorems between IOPVC and the Wolfe type dual model; In Sect. 4, we study the weak, strong, converse, restricted converse, and strict converse duality theorems between IOPVC and the Mond-Weir type dual model. Some examples are given to illustrate our conclusions.

Preliminary
Let X be a finite-dimensional Euclidean space. The notation ·, · denotes the inner product in X. For a pointx ∈ X, B(x; δ) := {x ∈ X : x -x < δ} denotes the open ball of radius δ aroundx. For a set C ⊂ X, span C, cone C stand for the linear hull and convex cone of C, respectively. Let C = ∅, the contingent cone of set C at the point x is defined by Let D be the set of all closed intervals in R. For any A = [a 1 , a 2 ] ∈ D, B = [b 1 , b 2 ] ∈ D, one has (see Moore [16]) where k is any real number. A partial ordering for intervals can be formulated as follows: Let F : X → D be a mapping on X defined as where F L , F U are the locally Lipschitz functions on X with F L (x) ≤ F U (x). Now, we consider the following semi-infinite interval-valued optimization problem with vanishing constraints(IOPVC): Let R |J| + denote the collection of all the functions λ : J → R taking values λ j > 0 only at finitely many points of J and equal to zero at the other points. For anyx ∈ E, I g (x) := {j ∈ J : g j (x) = 0} signifies the index set of all active constraints atx, and κ(x) := {λ j ∈ R |J| + : λ j g j (x) = 0, ∀j ∈ J} signifies the active constraint multipliers atx.
We give the following definitions of optimal solutions of (IOPVC).
(ii)x is said to be a locally weakly LU optimal solution of (IOPVC) if there exists an open ball B(x; δ) such that there is no x ∈ E ∩ B(x; δ) satisfying Let f : X → R be a locally Lipschitz function aroundx. The Clarke directional derivative of f aroundx in the direction v ∈ X and the Clarke subdifferential of f atx are, respectively, given by (see Clarke [17]) (ii) f is said to be strictly ∂ c -pseudoconvex atx if, for each x ∈ X, x =x and any The following sets of indicators, which will be used in the sequel, are given. Let x ∈ E, Referring to Definition 4 in [4], we give the following definition.
(i) The Abadie constraint qualification (ACQ) is said to hold atx iff T(E,x) = L(x), where L(x) is the linearized cone of (IOPVC) atx, and is the corresponding VC-linearized cone of (IOPVC) atx, and Remark 2.1 If the functions g j , h k , H i , G i are continuously differentiable, then the linearized cone and VC-linearized cone given in Definition 2.3 are the same as the linearized cones given in [4]. Now, we give the following theorem, the proof of which is similar to Proposition 1(ii) in [4].

Theorem 2.1
Letx ∈ E be a locally weakly LU optimal solution of (IOPVC) such that (VC-ACQ) holds atx and (2.4)

Definition 2.4 ([4])
The point x is said to be a VC-stationary point of (IOPVC) if there exist Lagrange multipliers α L , α U ∈ R + , λ g ∈ κ(x), λ h ∈ R n , λ H , λ G ∈ R l such that (2.3) and (2.4) hold. Now, let x be a VC-stationary point of (IOPVC) with corresponding multipliers λ g ∈ R |J| + , λ h ∈ R n , λ H , λ G ∈ R l , we give the following index sets:

Wolfe type duality
In this section, we refer to [6] to give the following Wolfe type dual models. First of all, let is an interval-valued function, and Now, we give the Wolfe type dual model of (IOPVC). For x ∈ E, In order to be independent of (IOPVC), we give another Wolfe type dual model: where E W denotes the set of all feasible points of (D W ) and prE W denotes the projection of the set E W on X.
Proof The proof is similar to Theorem 3.1.

Theorem 3.3 (Strong duality)
Letx ∈ E be a locally weakly LU optimal solution of (IOPVC) such that the (VC-ACQ) holds atx and is closed. Then there exist Lagrange mul- is a locally weakly LU optimal solution of (D W (x)).

Theorem 3.4 (Converse duality) Let x ∈ E be any feasible point of (IOPVC), and let
(3.6) If one of the following conditions holds: Thenȳ is the locally weakly LU optimal solution of (IOPVC).

by (3.3) and (3.6), one has
and by the definition of the index sets above, we get (3.10) By the ∂ c -quasiconvexity of the functions in assumption(ii) and (3.10), it follows that By the above inequality and 0 ∈ (ȳ), there exist ξ L ∈ ∂ c F L (ȳ) and ξ U ∈ ∂ c F U (ȳ) such that By (3.7) and the ∂ c -pseudoconvexity of F L (·) and F U (·), it follows that (3.11), so the result also holds.

thenx is the locally weakly LU optimal solution of (IOPVC).
Proof Suppose thatx is not a locally weakly LU optimal solution of (IOPVC), then there So,x is the locally weakly LU optimal solution of (IOPVC).
The following example shows that the conclusion of Theorem 3.5 holds.
Example 3.1 Let X = R 2 , n = 0, J = l = 1, consider the following question: The feasible set of problem (IOPVC1) is given by For any x ∈ E 1 , the Wolfe type dual model to (IOPVC1) is given by . Therefore, we can get the feasible set of problem (D W ), which is not dependent on x, we get x = (0, 0) is the locally weakly LU optimal solution of (IOPVC1). Theorem 3.6 (Strict converse duality) Letx ∈ E be the locally weakly LU optimal solution of (IOPVC) such that the (VC-ACQ) holds atx and is closed. Assume that the conditions of Theorem 3.3 hold and (ȳ, α L , α U , λ g , λ h , λ H , λ G , , v) ∈ E W (x) is the locally weakly LU optimal solution of (D W (x)).

Mond-Weir type duality
In this section, we give the Mond-Weir type dual model of (IOPVC) by referring to the new Mond-Weir type dual model(VC -MWD(x)) in [6]: for x ∈ E, In order to be independent of (IOPVC), we give the another Mond-Weir type dual model: where E MW denotes the set of all feasible points of (D MW ) and prE MW denotes the projection of the set E MW on X. .
, v) ∈ E MW be feasible points for the (IOPVC) and the (D MW ), respectively. If one of the following conditions holds: Proof Suppose that F(x) < s LU F(y), there exists and by the ∂ c -quasiconvexity of the above functions, one has Using the above inequality and 0 ∈ (y), there exist ξ L ∈ ∂ c F L (y) and ξ U ∈ ∂ c F U (y) such that By (4.2) and the ∂ c -pseudoconvexity of F L (·) and F U (·), it follows that Then α L ξ L + α U ξ U , xy < 0, α L , α U ∈ R + , α L + α U = 1, which contradicts (4.3).

Theorem 4.4 (Converse duality)
Let x ∈ E be any feasible point of (IOPVC) and (ȳ, α L , α U , λ g , λ h , λ H , λ G , , v) ∈ E MW be a feasible point of (D MW ). If one of the following conditions holds: Thenȳ is the locally weakly LU optimal solution of (IOPVC).
Proof Suppose to the contrary thatȳ is not the locally weakly LU optimal solution of (IOPVC), then one has (3.7).
(ii) The proof of (ii) is similar to the proof of Theorem 3.4(ii), so it is omitted.

Concluding remarks
In this paper, we study the duality theorems of nondifferentiable semi-infinite intervalvalued optimization problem with vanishing constraints. The weak duality, strong duality, converse duality, restricted converse duality, and strict converse duality theorems between (IOPVC) and its corresponding Wolfe and Mond-Weir type dual models are given under the conditions of ∂ c -pseudoconvex, strictly ∂ c -pseudoconvex, and ∂ c -quasiconvex.