Duality results for interval-valued semiinfinite optimization problems with equilibrium constraints using convexificators

Abstract

This paper deals with the study of interval-valued semiinfinite optimization problems with equilibrium constraints (ISOPEC) using convexificators. First, we formulate Wolfe-type dual problem for (ISOPEC) and establish duality results between the (ISOPEC) and the corresponding Wolfe-type dual under the assumption of $\partial ^{*}$-convexity. Second, we formulate Mond–Weir-type dual problem and propose duality results between the (ISOPEC) and the corresponding Mond–Weir-type dual under the assumption of $\partial ^{*}$-convexity, $\partial ^{*}$-pseudoconvexity, and $\partial ^{*}$-quasiconvexity.

1 Introduction

Due to a wide range of applications, the optimization problem with equilibrium constraints has been studied widely by many researchers. This problem regularly emerges in real-world applications, such as engineering designs, robust optimization, image restoration issues, fuzzy sets, and molecular distance geometry problems; see, for example, [1–8]. Optimization with equilibrium is a constrained optimization problem, with certain variational inequality or complementary constraints. The mathematical problem with equilibrium constraints (MPEC) belongs to a class of extremely significant problems. For a survey of recent developments and applications of optimization problems with equilibrium constraints, see [6, 9–15].

In nonsmooth optimization, the concept of convexificators has been shown to be a useful tool for deriving optimality and duality results. In 1994, Demyanov [16] developed the notion of a convexificator. A closed nonconvex convexificator for extended real-valued functions was developed by Jeyakumar and Luc [17]. In terms of convexificators, Wang and Jeyakumar [18] proposed the optimality conditions for optimization problems with equality constraints.

Throughout the last several decades, the study of dual problems connected to MPEC has been utilized to solve optimization issues in various fields, such as fractional problems, complex minimax problems, semiinfinite problems, and so on. Wolfe [19] and Mond–Weir [20] developed dual models for nonlinear programming problems. Rockafellar [21] used a conjugate function to study a fundamental duality theory for convex programming and developed an extended version of Fenchel’s duality theorem. Sun et al. [22] investigated Wolfe-type robust duality between the uncertain convex optimization problem and its uncertain dual problem by proving duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem. Sun et al. [23] introduced a mixed-type robust dual problem of this class of uncertain optimization problems and explored robust strong duality relations between them. Further, Sun et al. [24] established necessary and sufficient optimality conditions for robust approximate optimal solutions of this uncertain convex optimization problem, introduced a Wolfe-type robust approximate dual problem, and investigated robust approximate duality relations between them. The duality theory of nonlinear programming problems has received a lot of attention, see Mangasarian [3] and Mishra and Giorgi [25].

Semiinfinite programming is an optimization problem with an infinite number of inequality constraints on a feasible set. Semiinfinite optimization has recently become an active field in applied mathematics. Mishra et al. [26] established duality results for nonsmooth semiinfinite programming problems, and these results were extended for vanishing and equilibrium constraints by Pandey and Mishra [6, 7].

With the advancement of mathematics, an increasing number of researchers are focusing on interval-valued optimization problems. For continuously differentiable interval-valued optimization problems, Wu [27] considered the Wolfe-type dual problem. Tung [28] gave the optimality conditions and duality results for convex semiinfinite multiobjective interval-valued optimization problems. Using the concept of contingent epiderivatives, Su and Dinh [29] investigated the duality for interval-valued pseudoconvex optimization problems with equilibrium constraints. Recently, Wang and Wang [30] established duality theorems for nondifferentiable semiinfinite interval-valued optimization problems with vanishing constraints.

In this work, we consider the duality results for the interval-valued semiinfinite optimization problem with equilibrium constraints ((ISOPEC)) and show the dual relationships between (ISOPEC) and the corresponding Wolfe- and Mond–Weir-type dual models, using convexificators, which are inspired by the literature listed above. The organization of this paper is as follows. In Sect. 1, we recall some needful preliminaries and basic results. In Sect. 2, we establish weak and strong duality results between the (ISOPEC) and the corresponding Wolfe-type dual, using convexificators under the assumption of $\partial ^{*}$-convexity. In Sect. 3, we establish weak and strong duality results between the (ISOPEC) and the corresponding Mond–Weir-type dual, using convexificators under the assumption of $\partial ^{*}$-convexity, and in the last Sect. 4, we establish weak and strong duality results between the (ISOPEC) and the corresponding Mond–Weir-type dual, using convexificators under the assumption of $\partial ^{*}$-pseudoconvexity and $\partial ^{*}$-quasiconvexity.

2 Preliminaries

We recall some basic definitions and results in this section, which will be used in this paper.

We denote a finite-dimensional Euclidean space by X. The notation $\langle \cdot ,\cdot \rangle$ refers to the inner product. The symbol $B(\bar{z},\delta ):= \{{z\in X: \| z-\bar{z}\|} < \delta \}$ denotes the open ball of radius δ centered at z̄. Let C be a nonempty subset of X. We write $\mathrm{co}\ C$, $\mathrm{cone}\ C$, and $\mathrm{cl}\ C$ to denote the convex hull, convex cone, and closure of C, respectively, generated by C. Let $C\neq \emptyset$, then the contingent cone of set C at point z̄ is defined by

$$ T(C,\bar{z}) = \{v\in X \ \| \ \exists \ t_{n}\rightarrow 0 , \exists \ v_{n}\rightarrow \ v\ \text{such that} \ \bar{z} + t_{n}v_{n} \in C \ \forall n \in \mathbb{N} \}.$$

Let $\mathbb{D}$ be the set of all closed and bounded intervals in $\mathbb{R}$. For any $C= [c_{1}, c_{2}] \in \mathbb{D}$, $D = [d_{1}, d_{2}] \in \mathbb{D}$, one can see [31] that

$$ \begin{aligned} &C + D=[c_{1}+ d_{1},c_{2}+ d_{2}], \qquad -D = [-d_{2}, -d_{1}], \\ &C - D =[c_{1}- d_{2},c_{2}+ d_{1}], \qquad C+ k = [c_{1}+ k, c_{2}+ k], \end{aligned} $$

where k denotes a real number. A partial ordering for an interval can be expressed as follows:

$$\begin{aligned}& C \leq _{\mathrm{LU}} D \iff c_{1}\leq d_{1}, c_{2} \leq d_{2}, \\& C < _{\mathrm{LU}} D \iff C \leq _{\mathrm{LU}} D, C \neq D, \\& C\nless _{\mathrm{LU}} D\ \text{is the negative of} \ C< _{\mathrm{LU}} D, \\& C < _{\mathrm{LU}}^{s} D \iff c_{1}< d_{1}, c_{2} < d_{2}, \\& C\nless _{\mathrm{LU}}^{s} \ D \ \text{is the negative of}\ C< _{\mathrm{LU}}^{s} D. \end{aligned}$$

Let $F: X \rightarrow \mathbb{D}$ be a function on X, defined as

$$ F(z) = \bigl[F^{L}(z) ,F^{U}(z)\bigr] \quad \forall \ z \in X, $$

where $F^{L}$ and $F^{U}$ are functions defined on X with $F^{L}(z)\leq F^{U}(z)$, $\forall z\in X$. We consider the interval-valued semiinfinite optimization program with equilibrium constraints (ISOPEC)

$$\begin{aligned} &(\mathit{ISOPEC}) \quad \min F(z) = \bigl[F^{L}(z), F^{U}(z)\bigr] \\ &\quad \text{subject to}\quad g(z,t_{i}) \leq 0 \quad \forall t \in T,\ i= 1, \dots ,k, \\ &\hphantom{\quad \text{subject to}\quad } h_{j}(z)=0,\quad j= 1,\dots ,p, \\ &\hphantom{\quad \text{subject to}\quad } G_{i}(z) \geq 0,\quad i=1,\dots ,l, \\ &\hphantom{\quad \text{subject to}\quad } H_{i}(z)\geq 0,\quad i=1,\dots ,l, \\ &\hphantom{\quad \text{subject to}\quad } G_{i}(z)^{T}H_{i}(z)= 0, \quad i=1,\dots,l, \end{aligned}$$

(1)

where the index set T is an arbitrary nonempty set, not necessarily finite; functions $g: X \times T \rightarrow \mathbb{R}$, $h_{j}: X \rightarrow \mathbb{R}$, $G_{i}: X \rightarrow \mathbb{R}\mathbbm{,} \ \text{and} \ H_{i}: X \rightarrow \mathbb{R}$ are locally Lipschitz on X and $(\cdot )^{T}$ indicates the transpose. The feasible set of problem (ISOPEC) is

$$\begin{aligned} M:={}& \bigl\{ z\in X : g(z,t_{i}) \leq 0 \ \forall t \in T, i= 1, \dots ,k, h_{j}(z)=0, j= 1,\dots ,p, \\ &G_{i}(z) \geq 0, i=1,\dots ,l, H_{i}(z)\geq 0, i=1,\dots ,l, G_{i}(z)^{T}H_{i}(z)= 0, \\ &i=1,\dots ,l\bigr\} . \end{aligned}$$

(2)

Definition 2.1

Suppose $\bar{z} \in M$.

• 1. A feasible point z̄ is said to be a locally LU-optimal solution of (ISOPEC) if there exists a neighborhood $N(\bar{z})$ of z̄ such that there is no $z \in M \cap N(\bar{z})$, for which

  $$ F(z)< _{\mathrm{LU}} F(\bar{z}).$$

• 2. A feasible point z̄ is said to be a locally weakly LU-optimal solution of (ISOPEC) if there exists a neighborhood $N(\bar{z})$ of z̄ such that there is no $z \in M \cap N(\bar{z})$, for which

  $$ F(z)< _{\mathrm{LU}}^{s} F(\bar{z}).$$

Let $\bar{z} \in M$ be a feasible vector for (ISOPEC). We define the following index sets to be used in the sequel:

$$\begin{aligned}& T_{g}:= T_{g}(\bar{z}):= \bigl\{ t\in T\, :\, g( \bar{z},t)=0\bigr\} , \\& \alpha := \alpha (\bar{z})= \bigl\{ i =1,2,\dots ,l : \ G_{i}( \bar{z})=0, H_{i}( \bar{z})>0\bigr\} , \\& \beta := \beta (\bar{z})= \bigl\{ i =1,2,\dots ,l :\ G_{i}( \bar{z})=0, H_{i}( \bar{z})=0\bigr\} , \\& \gamma := \gamma (\bar{z})= \bigl\{ i =1,2,\dots ,l : \ G_{i}( \bar{z})>0, H_{i}( \bar{z})=0\bigr\} , \\& \nu _{1} = \alpha \cup \beta , \qquad \nu _{2} = \gamma \cup \beta . \end{aligned}$$

The set β is called a degenerate set. The vector z̄ is said to satisfy the strict complementarity condition if β is empty.

Let $f: X \rightarrow \mathbb{R}\cup \{+\infty \}$ be an extended real-valued function and $z\in X$ be such that $f(z)$ is finite. Then the lower and upper Dini directional derivatives of f at z in the direction d, respectively, are given as

$$\begin{aligned} f_{d}^{-}(x,d) &= \liminf \limits _{x\rightarrow 0^{+}} \ \frac{f(z+td)-f(z)}{t}, \\ f_{d}^{+}(x,d) &= \limsup \limits _{x\rightarrow 0^{+}} \ \frac{f(z+td)-f(z)}{t}. \end{aligned}$$

Definition 2.2

([17])

A function $f: X \rightarrow \mathbb{R}\cup \{+\infty \}$ is said to admit an upper convexificator $\partial ^{*}f(z)$ at $z \in \ X$ if $\partial ^{*}f(z) \subset X$ is a closed set and, for every $d \in X$,

$$ f_{d}^{+}(x,d) \leq \limsup \limits _{\xi \in \partial ^{*}f(z) }\ \langle \xi , d \rangle .$$

The function f is said to admit a lower convexificator $\partial ^{*}f(z)$ at $z \in \ X$ if $\partial ^{*}f(z) \subset X$ is a closed set and, for every $d \in X$,

$$ f_{d}^{-}(z,d) \geq \liminf \limits _{\xi \in \partial ^{*}f(z) }\ \langle \xi , d \rangle .$$

A closed set $\partial ^{*}f(z)$ is said to be a convexificator of f at z if it is both an upper and lower convexificator.

Now, we collect the definition of upper and lower semiregular convexificator from [17, 32].

Definition 2.3

Let $f : X \rightarrow \mathbb{R} \cup \{+\infty \}$ be an extended real-valued function and let $z \in X$ be such that $f(z)$ is finite. The function f is said to admit an upper semiregular convexificator $\partial ^{*}f (z) \subset X$ at z if $\partial ^{*} f(z)$ is closed and, for each $d\in X$,

$$ f^{+}(z; d ) \leq \sup_{\xi \in \partial ^{*}f(z)}\langle \xi , d \rangle . $$

The function f is said to admit a lower semiregular convexificator $\partial _{*} f(z) \subset X$ at z if $\partial _{*} f(z)$ is closed and, for each $d \in X$,

$$ f^{-}(z; d ) \geq \inf_{\xi \in \partial _{*}f(z)} \langle \xi , d \rangle .$$

The set $\partial ^{*}f(z)$ is said to be a semiregular convexificator, if it is both an upper and lower semiregular convexificator.

Definition 2.4

([33])

Let $f: X \rightarrow \mathbb{R}\cup \{+\infty \}$ be an extended real-valued function and suppose it admits an upper semiregular convexificator at $z \in X$. Then the function f is said to be $\partial ^{*}$-convex at x̄ iff

$$ f(z)\geq f(\bar{z}) + \langle \xi , z-\bar{z}\rangle , \quad \forall \ z\in X, \forall \ \xi \in \partial ^{*}f(\bar{z}).$$

Example 2.1

Consider the function $f:\mathbb{R}^{2}\rightarrow \mathbb{R}$ given by

$$ f(z) = \lvert z_{1} \rvert + \lvert z_{2} \rvert ,\quad \text{where}\ z=(z_{1},z_{2}). $$

The set $\partial ^{*} f(0,0)=\{(-1,1),(-1,-1),(1,1), (1,-1)\}$ is an upper semiregular convexificator of $f(z)$ at $\bar{z}=(0,0)$. Then,

$$ f(z)-f(\bar{z})=z_{1}^{2}+z_{2}^{2} \geq \xi _{1}z_{1}+\xi _{2}z_{2}= \langle \xi , z-\bar{z}\rangle , \quad \forall \ \xi \in \mathrm{co} \,\partial _{*} f(\bar{z}). $$

Thus, we conclude that f is $\partial ^{*}$-convex.

Definition 2.5

([34])

Let $f: X \rightarrow \mathbb{R}\cup \{+\infty \}$ be an extended real-valued function and suppose it admits an upper semiregular convexificator at $z \in X$. Then the function f is said to be $\partial ^{*}$-pseudoconvex at x̄ iff

$$ \forall z\in X \quad \text{with } z\neq \bar{z}, f(z)< f(\bar{z}) \quad \implies\quad \langle \xi , z-\bar{z}\rangle < 0, \quad \forall z\in X, \forall \xi \in \partial ^{*}f(\bar{z}).$$

Example 2.2

([17])

Consider the function $f:\mathbb{R}\rightarrow \mathbb{R}$ given by

$$ f(z) = \textstyle\begin{cases} \sqrt{z}, &z\geq 0, \\ -\sqrt{-z}, &z< 0. \end{cases} $$

The set $\partial ^{*} f(0)=[1, +\infty )$ is an upper convexificator of f at 0 and

$$\begin{aligned} \partial ^{*} f(z)&=\frac{1}{2\sqrt{z}},\quad z>0. \\ \partial ^{*} f(z)&=\frac{1}{2\sqrt{-z}},\quad z< 0. \end{aligned}$$

Also f is $\partial ^{*}$-pseudoconvex on $\mathbb{R}$, but not $\partial ^{*}$-convex.

We extend the following sets of indicators and definitions for interval-valued semiinfinite mathematical programming problem with equilibrium constraints (ISOPEC), which will be used in the sequel, that were given for semiinfinite mathematical programming problem with equilibrium constraints (SIMPEC) by Pandey and Mishra [7]:

$$ \begin{gathered} g= \bigcup_{i=1}^{k} \mathrm{co}\,\partial ^{*}g(\bar{z},t_{i}), \qquad h= \bigcup_{i=1}^{p} \mathrm{co}\,\partial ^{*}h_{i}(\bar{z}) \cup \mathrm{co}\,\partial ^{*}(-h_{i}) (\bar{z}), \\ G_{\alpha}= \bigcup_{i\in \alpha} \mathrm{co}\, \partial ^{*}G_{i}( \bar{z}) \cup \mathrm{co}\,\partial ^{*}(-G_{i}) (\bar{z}),\qquad G_{\beta}= \bigcup _{i\in \beta} \mathrm{co}\,\partial ^{*}G_{i}( \bar{z}), \\ H_{\gamma}= \bigcup_{i\in \gamma} \mathrm{co}\, \partial ^{*}H_{i}( \bar{z}) \cup \mathrm{co}\,\partial ^{*}(-H_{i}) (\bar{z}), \qquad H_{\beta}= \bigcup _{i\in \beta} \mathrm{co}\,\partial ^{*}H_{i}( \bar{z}), \\ (\mathit{GH})_{\beta}= \bigcup_{i\in \beta} \mathrm{co}\,\partial ^{*}(-G_{i}) ( \bar{z})\cup \mathrm{co}\,\partial ^{*}(-H_{i}) (\bar{z}), \\ \Gamma (\bar{z}):= g^{-}\cap h^{-}\cap G_{\alpha}^{-}\cap H_{\gamma}^{-} \cap ( \mathit{GH})_{\beta}^{-}, \end{gathered} $$

where $t_{1},t_{2},\dots ,t_{k}\in T_{g}(\bar{z})\leq n+1$.

Definition 2.6

If all functions have upper convexificators at a feasible point z̄ of (ISOPEC), we say that the generalized standard Abadie constraints condition (GS-ACQ) is satisfied at z̄ if at least one of the dual sets used in the definition of $\Gamma (\bar{z})$ is nonempty and

$$ \Gamma (\bar{z})\subseteq T(C,\bar{z}).$$

Definition 2.7

(Generalized alternatively stationary point)

Let z̄ be a feasible point of (SIOPEC). A generalized alternatively stationary point, denoted by GA-stationary, is said to be at z̄ if there exist $\alpha ^{L},\alpha ^{U} \in \mathbb{R}$, $\eta = (\eta ^{g}, \eta ^{h}, \eta ^{G}, \bar{\eta}^{H})\in \mathbb{R}^{k+p+2l}$, $\mu = (\mu ^{h}, \mu ^{G}, \mu ^{H})\in \mathbb{R}^{p+2l}$, and indices $t_{1},t_{2},\dots ,t_{k} \in T_{g}(\bar{z})$, $k\leq n+1$ such that the following conditions hold:

$$\begin{aligned}& \begin{aligned}[b] 0\in{}& \alpha ^{L}\ \mathrm{co}\,\partial ^{*}F^{L}(w) + \alpha ^{U} \mathrm{co}\,\partial ^{*}F^{U}(w) + \sum_{i=1}^{k}\eta _{i}^{g} \mathrm{co}\,\partial ^{*}g(w,t_{i}) \\ &{} + \sum_{j=1}^{p}\bigl[\eta _{j}^{h}\mathrm{co}\,\partial ^{*}h_{j}(w)+ \mu _{j}^{h}\mathrm{co}\,\partial ^{*}(-h_{j}) (w)\bigr] \\ &{} + \sum_{i=1}^{l}\bigl[\eta _{i}^{G}\mathrm{co}\,\partial ^{*}(-G)_{i}(w)+ \eta _{i}^{H}\mathrm{co}\,\partial ^{*}(-H_{i}) (w)\bigr] \\ &{} +\sum_{i=1}^{l}\bigl[\mu _{i}^{G}\mathrm{co}\,\partial ^{*}(G)_{i}(w)+ \mu _{i}^{H}\mathrm{co}\,\partial ^{*}(H_{i}) (w)\bigr], \end{aligned} \end{aligned}$$

(3)

$$\begin{aligned}& \begin{gathered} \alpha ^{L}, \alpha ^{U} \in [0,1],\quad \alpha ^{L}+ \alpha ^{U}= 1, \qquad \eta _{I_{g}}^{g} \geq 0,\\ \eta _{j}^{h}\geq 0, \qquad \mu _{j}^{h}\geq 0, \quad j=1,\dots ,p, \end{gathered} \end{aligned}$$

(4)

$$\begin{aligned}& \eta _{i}^{G}\geq 0,\qquad \eta _{i}^{H} \geq 0,\qquad \mu _{i}^{G}\geq 0,\qquad \mu _{i}^{H} \geq 0, \quad i= 1,\dots ,l, \end{aligned}$$

(5)

$$\begin{aligned}& \eta _{\gamma}^{G} = \eta _{\alpha}^{H} = \mu _{\gamma}^{G} = \mu _{ \alpha}^{H} = 0 ,\quad \forall i \in \beta , \qquad \mu _{i}^{G} = 0 \quad \text{or} \quad \mu _{i}^{H} = 0. \end{aligned}$$

(6)

Definition 2.8

(Generalized strong stationary point)

Let z̄ be a feasible point of (ISOPEC). A generalized alternatively stationary point, denoted by GA-stationary, is said to be at z̄ if there exist $\alpha ^{L},\alpha ^{U} \in \mathbb{R}$ and $\eta = (\eta ^{g}, \eta ^{h}, \eta ^{G}, \eta ^{H})\in \mathbb{R}^{k+p+2l}$, $\ \mu = (\mu ^{h}, \mu ^{G}, \mu ^{H})\in \mathbb{R}^{p+2l}$, and indices $t_{1},t_{2},\dots ,t_{k} \in T_{g}(\bar{z})$, $k\leq n+1$, satisfying conditions (3)–(5) together with the following conditions:

$$\begin{aligned} \eta _{\gamma}^{G} = \eta _{\alpha}^{H} = \mu _{\gamma}^{G} = \mu _{ \alpha}^{H} = 0 ,\quad \forall \ i \in \beta , \ \mu _{i}^{G} = 0, \ \mu _{i}^{H} = 0. \end{aligned}$$

3 Duality

In this section we formulate Wolfe- and Mond–Weir-type dual model for (ISOPEC) and establish the weak and strong duality results in terms of convexificators. Let $\bar{z} \in M$ be any feasible solution of (ISOPEC). The following index sets will be used in the sequel:

$$\begin{aligned} \gamma _{\mu}^{+}&:= \bigl\{ i\in \gamma : \mu _{i}^{H}>0\bigr\} ; \\ \alpha _{\mu}^{+}&:= \bigl\{ i\in \alpha : \mu _{i}^{G}>0\bigr\} ; \\ \beta _{\mu}^{G}&:= \bigl\{ i\in \beta : \mu _{i}^{H}=0,\,\mu _{i}^{G}>0 \bigr\} ; \\ \beta _{\mu}^{H}&:= \bigl\{ i\in \beta : \mu _{i}^{G}=0,\,\mu _{i}^{H}>0 \bigr\} ; \\ L_{\mu}&=\gamma _{\mu}^{+}\cup \alpha _{\mu}^{+}\cup \beta _{\mu}^{G} \cup \beta _{\mu}^{H}. \end{aligned}$$

Definition 3.1

(Wolfe dual)

The Wolfe dual (WDISOPEC) for the interval-valued semiinfinite optimization problem with equilibrium constraints (ISOPEC) is defined by

$$\begin{aligned} &\max_{w,\alpha ,\eta} \Biggl[F(w)+\sum _{i=1}^{k}\eta _{i}^{g}g(w,t_{i}) +\sum_{j=1}^{p}\bigl(\eta _{j}^{h}-\mu _{j}^{h} \bigr)h_{j}(w)-\sum_{i=1}^{l} \bigl(\eta _{i}^{G}G_{i}(w)+ \eta _{i}^{H}H_{i}(w)\bigr)\Biggr] \\ &\quad \text{subject to} \quad 0\in \alpha ^{L}\mathrm{co}\,\partial ^{*}F^{L}(w) + \alpha ^{U}\mathrm{co}\, \partial ^{*}F^{U}(w) + \sum_{i=1}^{k} \eta _{i}^{g}\mathrm{co}\,\partial ^{*}g(w,t_{i}) \\ &\quad\hphantom{\text{subject to} \quad 0\in }{} + \sum_{j=1}^{p}\bigl[\eta _{j}^{h}\mathrm{co}\,\partial ^{*}h_{j}(w)+ \mu _{j}^{h}\mathrm{co}\,\partial ^{*}(-h_{j}) (w)\bigr] \\ &\quad\hphantom{\text{subject to} \quad 0\in }{} + \sum_{i=1}^{l}\bigl[\eta _{i}^{G}\mathrm{co}\,\partial ^{*}(-G)_{i}(w)+ \eta _{i}^{H}\mathrm{co}\,\partial ^{*}(-H_{i}) (w)\bigr], \\ &\alpha ^{L}, \alpha ^{U}\in (0,1), \quad \alpha ^{L}+ \alpha ^{U}= 1, \qquad \eta _{T_{g}}^{g} \geq 0, \qquad \eta _{j}^{h}\geq 0, \qquad \mu _{j}^{h} \geq 0, \quad j=1,\dots ,p, \\ &\eta _{i}^{G}\geq 0,\qquad \eta _{i}^{H} \geq 0,\qquad \mu _{i}^{G}\geq 0,\qquad \mu _{i}^{H} \geq 0, \quad i= 1,\dots ,l, \\ &\eta _{\gamma}^{G} = \eta _{\alpha}^{H} = \mu _{\gamma}^{G} = \mu _{ \alpha}^{H} = 0 ,\quad \forall i \in \beta , \quad \mu _{i}^{G} = 0, \qquad \mu _{i}^{H} = 0, \end{aligned}$$

(7)

where $\eta = (\eta ^{g}, \eta ^{h}, \eta ^{G}, \eta ^{H}) \in \mathbb{R}^{k+p+2l}$, $\mu \in (\mu ^{h},\mu ^{G},\mu ^{H}) \in \mathbb{R}^{p+2l}$ and $t_{1},t_{2},\dots ,t_{k} \in T_{g}(\bar{z})$, $k \leq n+1$.

Theorem 3.2

(Weak duality)

Let z̄ and $(w,\alpha ^{L},\alpha ^{U},\eta )$ be feasible points for the interval-valued semiinfinite optimization problem (ISOPEC) and the dual problem (WDISOPEC), respectively, and let the index set $T_{g}$, as well as α, β, γ, be defined accordingly. Suppose the functions $F^{L};\, F^{U}; \, g(\cdot ,t)$, for $t\in T$; $\pm h_{j}$, for $j=1,\dots ,p;\, -G_{i}$, for $i\in \nu _{1}$, and $-H_{i}$, for $i\in \nu _{2}$, admit bounded upper semiregular convexificators and $\partial ^{*}$-convex functions at w. If $L_{\mu} = \phi$, then

$$ \begin{aligned}[b] F(z) \nless ^{s}_{\mathrm{LU}}{}& F(w)+\sum\limits _{i=1}^{k}\eta _{i}^{g}g(w,t_{i}) +\sum\limits _{j=1}^{p}\bigl(\eta _{j}^{h}-\mu _{j}^{h} \bigr)h_{j}(w) \\ &{}-\sum\limits _{i=1}^{l}\bigl(\eta _{i}^{G}G_{i}(w)+ \eta _{i}^{H}H_{i}(w)\bigr). \end{aligned} $$

(8)

Proof

Based on the initial hypothesis, we have the following system:

$$\begin{aligned}& g(z,t_{i}) \leq 0, \quad \forall t \in T, \end{aligned}$$

(9)

$$\begin{aligned}& h_{j}(z)= 0,\quad \forall j=1,\dots ,p, \end{aligned}$$

(10)

$$\begin{aligned}& -G_{i}(z) \leq 0,\quad \forall i=1,\dots ,l, \end{aligned}$$

(11)

$$\begin{aligned}& -H_{i}(z) \leq 0,\quad \forall i=1,\dots ,l. \end{aligned}$$

(12)

Since $F^{L}$ and $F^{U}$ are convex functions,

$$\begin{aligned} &F^{L}(z)-F^{L}(w)\geq \langle \xi _{1},z-w \rangle , \quad \forall \xi _{1} \in \partial ^{*}F^{L}(w), \end{aligned}$$

(13)

$$\begin{aligned} &F^{U}(z)-F^{U}(w)\geq \langle \xi _{2},z-w \rangle , \quad \forall \xi _{2} \in \partial ^{*}F^{U}(w). \end{aligned}$$

(14)

Similarly, we have

$$\begin{aligned}& g(z,t_{i})-g(w,t_{i})\geq \bigl\langle \xi _{i}^{g},z-w \bigr\rangle , \quad \forall \xi _{i}^{g} \in \partial ^{*}g(w,t_{i}), \forall t_{i} \in T_{g}(\bar{z}), \end{aligned}$$

(15)

$$\begin{aligned}& h_{j}(z)- h_{j}(w) \geq \bigl\langle \zeta _{j}^{h_{1}},z-w \bigr\rangle , \quad \forall \zeta _{j}^{h_{1}} \in \partial ^{*}h_{j}(w), \forall j={1,\dots ,p}, \end{aligned}$$

(16)

$$\begin{aligned}& -h_{j}(z)+ h_{j}(w) \geq \bigl\langle \zeta _{j}^{h_{2}},z-w \bigr\rangle , \quad \forall \zeta _{j}^{h_{2}} \in \partial ^{*}(-h_{j}) (w), \forall j=1,\dots ,p, \end{aligned}$$

(17)

$$\begin{aligned}& -G_{i}(z)+ G_{i}(w) \geq \bigl\langle \xi _{i}^{G},z-w \bigr\rangle , \quad \forall \xi _{i}^{G} \in \partial ^{*}(-G_{i}) (w), \forall i\in \nu _{1}, \end{aligned}$$

(18)

$$\begin{aligned}& -H_{i}(z)+ H_{i}(w) \geq \bigl\langle \xi _{i}^{H},z-w \bigr\rangle , \quad \forall \xi _{i}^{H} \in \partial ^{*}(-H_{i}) (w), \forall i\in \nu _{2}. \end{aligned}$$

(19)

If $L_{\mu} = \phi$, multiplying (13)–(19) by $\alpha ^{L},\alpha ^{U},\eta _{i}^{g} \geq 0$, $t_{i}\in T_{g}$, $\eta _{j}^{h}> 0$, $j=1,\dots ,p$, $\mu _{j}^{h}> 0$, $j=1,\dots ,p$, $\eta _{i}^{G}> 0$, $i\in \nu _{1}$, $\eta _{i}^{H}> 0$, $i\in \nu _{2}$, respectively, and adding (13)–(19), we have

$$\begin{aligned} &\alpha ^{L}\bigl(F^{L}(z)-F^{L}(w) \bigr)+\alpha ^{U}\bigl(F^{U}(z) - F^{U}(w) \bigr) + \sum_{i=1}^{k}\eta _{i}^{g}g(z,t_{i}) - \sum _{i=1}^{k} \eta _{i}^{g} g(w,t_{i}) \\ &\qquad {}+ \sum_{j=1}^{p}\eta _{j}^{h}h_{j}(z) - \sum _{j=1}^{p} \eta _{j}^{h}h_{j}(w) - \sum_{j=1}^{p}\mu _{j}^{h}h_{j}(z) + \sum _{j=1}^{p}\mu _{j}^{h}h_{j}(w) \\ &\qquad {}-\sum_{i=1}^{l}\eta _{i}^{G}G_{i}(z) + \sum _{i=1}^{l} \eta _{i}^{G}G_{i}(w) - \sum_{i=1}^{l}\eta _{i}^{H}H_{i}(z) + \sum _{i=1}^{l}\eta _{i}^{H}H_{i}(w) \\ &\quad \geq \Biggl\langle \xi _{\alpha} + \sum_{i=1}^{k} \eta _{i}^{g} \xi _{i}^{g} + \sum_{j=1}^{p}\bigl[\eta _{j}^{h}\zeta _{j}^{h_{1}}+ \mu _{j}^{h}\zeta _{j}^{h_{2}}\bigr] + \sum_{i=1}^{l}\bigl[\eta _{i}^{G} \xi _{i}^{G}+\eta _{i}^{H}\xi _{i}^{H}\bigr], z-w \Biggr\rangle , \end{aligned}$$

(20)

where $\xi _{\alpha}=\alpha ^{L}\xi _{1} + \alpha ^{U}\xi _{2}$.

From (7), there exist $\bar{\xi _{1}} \in \mathrm{co}\,\partial ^{*}F^{L}(w)$, $\ \bar{\xi _{2}} \in \mathrm{co}\,\partial ^{*}F^{U}(w)$, $\bar{\xi}_{i}^{g} \in \mathrm{co}\,\partial ^{*}g(w,t_{i})$, $\bar{\zeta}_{j}^{h_{1}} \in \mathrm{co}\,\partial ^{*}h_{j}(w)$, $\bar{\zeta}_{j}^{h_{2}}\in \mathrm{co}\,\partial ^{*}(-h_{j})(w)$, $\bar{\xi}_{i}^{G} \in \mathrm{co}\,\partial ^{*}(-G_{i})(w)$, and $\bar{\xi}_{i}^{H} \in \mathrm{co}\,\partial ^{*}(-H_{i})(w)$, such that

$$ \bar{\xi}_{\alpha} + \sum_{i=1}^{k} \eta _{i}^{g}\bar{\xi}_{i}^{g} + \sum_{j=1}^{p}\bigl[\eta _{j}^{h}\bar{\zeta}_{j}^{h_{1}}+ \mu _{j}^{h} \bar{\zeta}_{j}^{h_{2}} \bigr] + \sum_{i=1}^{l}\bigl[\eta _{i}^{G} \bar{\xi}_{i}^{G}+ \eta _{i}^{H}\bar{\xi}_{i}^{H} \bigr] = 0.$$

Therefore, it follows from (20) that

$$\begin{aligned} &\alpha ^{L}\bigl(F^{L}(z)- F^{L}(w)\bigr)+\alpha ^{U}\bigl(F^{U}(z) - F^{U}(w)\bigr) + \sum_{i=1}^{k} \eta _{i}^{g}g(z,t_{i}) - \sum _{i=1}^{k} \eta _{i}^{g} g(w,t_{i}) \\ &\quad {}+ \sum_{j=1}^{p}\eta _{j}^{h}h_{j}(z) - \sum _{j=1}^{p} \eta _{j}^{h}h_{j}(w) - \sum_{j=1}^{p}\mu _{j}^{h}h_{j}(z) + \sum _{j=1}^{p}\mu _{j}^{h}h_{j}(w) -\sum_{i=1}^{l} \eta _{i}^{G}G_{i}(z) \\ &\quad {}+ \sum_{i=1}^{l}\eta _{i}^{G}G_{i}(w) - \sum _{i=1}^{l} \eta _{i}^{H}H_{i}(z) + \sum_{i=1}^{l}\eta _{i}^{H}H_{i}(w) \geq 0. \end{aligned}$$

(21)

Now, combining (9)–(12), we get

$$\begin{aligned} \sum_{i=1}^{k}\eta _{i}^{g}g(z,t_{i}) + \sum _{j=1}^{p} \eta _{j}^{h}h_{j}(z) - \sum_{j=1}^{p}\mu _{j}^{h}h_{j}(z) - \sum _{i=1}^{l}\eta _{i}^{G}G_{i}(z) - \sum_{i=1}^{l} \eta _{i}^{H}H_{i}(z) \leq 0 . \end{aligned}$$

This, together with equation (21), leads to

$$ \begin{aligned}[b] \alpha ^{L}F^{L}(z)+ \alpha ^{U}F^{U}(z)\geq{}& \alpha ^{L}F^{U}(w)+ \alpha ^{U}F^{U}(w) + \sum\limits _{i=1}^{k}\eta _{i}^{g}g(w,t_{i}) \\ &{} +\sum\limits _{j=1}^{p}\bigl(\eta _{j}^{h}-\mu _{j}^{h} \bigr)h_{j}(w)- \sum\limits _{i=1}^{l} \bigl(\eta _{i}^{G}G_{i}(w)+ \eta _{i}^{H}H_{i}(w)\bigr). \end{aligned} $$

(22)

In fact, if (8) does not hold, then there would be

$$ F(z) < ^{s}_{\mathrm{LU}} F(w)+\sum_{i=1}^{k} \eta _{i}^{g}g(w,t_{i}) + \sum _{j=1}^{p}\bigl(\eta _{j}^{h}- \mu _{j}^{h}\bigr)h_{j}(w)-\sum _{i=1}^{l}\bigl(\eta _{i}^{G}G_{i}(w)+ \eta _{i}^{H}H_{i}(w)\bigr). $$

(23)

By definition, we have

$$ \textstyle\begin{cases} F^{L}(z) < F^{L}(w)+\sum _{i=1}^{k}\eta _{i}^{g}g(w,t_{i}) + \sum _{j=1}^{p}(\eta _{j}^{h}-\mu _{j}^{h})h_{j}(w) \\ \hphantom{F^{L}(z) < }{}-\sum _{i=1}^{l}(\eta _{i}^{G}G_{i}(w)+ \eta _{i}^{H}H_{i}(w)), \\ F^{U}(z) < F^{U}(w)+\sum _{i=1}^{k}\eta _{i}^{g}g(w,t_{i}) + \sum _{j=1}^{p}(\eta _{j}^{h}-\mu _{j}^{h})h_{j}(w) \\ \hphantom{F^{U}(z) < }{} -\sum _{i=1}^{l}(\eta _{i}^{G}G_{i}(w)+ \eta _{i}^{H}H_{i}(w)), \end{cases} $$

or

$$ \textstyle\begin{cases} F^{L}(z) < F^{L}(w)+\sum _{i=1}^{k}\eta _{i}^{g}g(w,t_{i}) + \sum _{j=1}^{p}(\eta _{j}^{h}-\mu _{j}^{h})h_{j}(w) \\ \hphantom{F^{L}(z) < }{} -\sum _{i=1}^{l}(\eta _{i}^{G}G_{i}(w)+ \eta _{i}^{H}H_{i}(w)), \\ F^{U}(z) \leq F^{U}(w)+\sum _{i=1}^{k}\eta _{i}^{g}g(w,t_{i}) +\sum _{j=1}^{p}(\eta _{j}^{h}-\mu _{j}^{h})h_{j}(w) \\ \hphantom{F^{U}(z) \leq}{} -\sum _{i=1}^{l}(\eta _{i}^{G}G_{i}(w)+ \eta _{i}^{H}H_{i}(w)), \end{cases} $$

or

$$ \textstyle\begin{cases} F^{L}(z) \leq F^{L}(w)+\sum _{i=1}^{k}\eta _{i}^{g}g(w,t_{i}) +\sum _{j=1}^{p}(\eta _{j}^{h}-\mu _{j}^{h})h_{j}(w) \\ \hphantom{F^{L}(z) \leq}{} -\sum _{i=1}^{l}(\eta _{i}^{G}G_{i}(w)+ \eta _{i}^{H}H_{i}(w)), \\ F^{U}(z) < F^{U}(w)+\sum _{i=1}^{k}\eta _{i}^{g}g(w,t_{i}) + \sum _{j=1}^{p}(\eta _{j}^{h}-\mu _{j}^{h})h_{j}(w) \\ \hphantom{F^{U}(z) < }{} -\sum _{i=1}^{l}(\eta _{i}^{G}G_{i}(w)+ \eta _{i}^{H}H_{i}(w)). \end{cases} $$

Since $\alpha ^{L},\alpha ^{U}\in (0,1)$ and $\alpha ^{L}+\alpha ^{U} =1$, the strict inequality holds:

$$ \begin{aligned}[b] \alpha ^{L}F^{L}(z)+ \alpha ^{U}F^{U}(z) < {}& \alpha ^{L}F^{L}(w)+ \alpha ^{U}F^{U}(w) + \sum\limits _{i=1}^{k}\eta _{i}^{g}g(w,t_{i}) \\ &{} +\sum\limits _{j=1}^{p}\bigl(\eta _{j}^{h}-\mu _{j}^{h} \bigr)h_{j}(w)- \sum\limits _{i=1}^{l} \bigl(\eta _{i}^{G}G_{i}(w)+ \eta _{i}^{H}H_{i}(w)\bigr), \end{aligned} $$

(24)

which contradicts (22) and hence the proof is complete. □

Example 3.1

Consider the following $(\mathit{WDISOPEC})$ problem in $\mathbb{R}^{2}$:

$$\begin{aligned} &\min \ F(z)=\bigl[F^{L}(z_{1},z_{2}), F^{U}(z_{1},z_{2})\bigr] = \biggl[ \frac{z_{1}^{2}}{2},\frac{z_{2}^{2}}{2} \biggr], \\ &\quad \text{subject to} \quad g(z ,t) =-z_{2}-t\leq 0,\quad \forall t\in [0,1], \\ &\hphantom{\quad \text{subject to} \quad }G_{1}(z) =-z_{1} \geq 0, \\ &\hphantom{\quad \text{subject to} \quad }H_{1}(z) =- \vert z_{1} \vert - \vert z_{2}+1 \vert +1 \geq 0, \\ &\hphantom{\quad \text{subject to} \quad }G_{1}(z)H_{1}(z )= -z_{1}\bigl(- \vert z_{1} \vert - \vert z_{2}+1 \vert +1 \bigr)=0. \end{aligned}$$

The feasible set, denoted by M, is defined by

$$\begin{aligned} M= \bigl\{ (z_{1},z_{2}) \in \mathbb{R}^{2} \ : z_{1}=0,z_{2}=0 \bigr\} . \end{aligned}$$

(25)

Clearly, $z^{*}= (0,0)$ is a feasible point. Then, $T_{g}(0,0)=\{0\}$, $\beta (0,0)=\{1\}$.

Now consider

$$\begin{aligned} (\mathit{WDISOPEC}) \bigl(z^{*}\bigr)\ \max \ F(w)+\eta _{1}^{g}g(w,0)-\eta _{1}^{G}G_{1}(w)- \eta _{1}^{H}H_{1}(w), \end{aligned}$$

(26)

subject to

$$ 0\in \mathrm{co}\,\partial ^{*}F(w)+\eta _{1}^{g} \mathrm{co}\, \partial ^{*}g(w,0)+\eta _{1}^{G} \mathrm{co}\,\partial ^{*}(-G_{1}) (w)+ \eta _{1}^{H}\mathrm{co}\,\partial ^{*}(-H_{1}) (w),$$

where

$$\begin{aligned}& \mathrm{co}\,\partial ^{*}F(w)=\alpha ^{L}(w_{1},0)+ \alpha ^{U}(0,w_{2}), \quad \alpha ^{L}, \alpha ^{U}\in (0,1), \alpha ^{L}+\alpha ^{U}=1, \\& \mathrm{co}\,\partial ^{*}g(w,0)=\mathrm{co}\,\bigl\{ (0,-1)\bigr\} , \qquad \mathrm{co}\,\partial ^{*}(-G_{1}) (w)=\mathrm{co}\, \bigl\{ (1,0)\bigr\} , \quad \text{and}\\& \mathrm{co}\,\partial ^{*}(-H_{1}) (w) = \textstyle\begin{cases} \mathrm{co}\,\{ (1,-1)\}, &\text{if}\ w_{1}>0, w_{2}< -1, \\ \mathrm{co}\,\{ (1,1),(1,-1)\}, &\text{if}\ w_{1}>0, w_{2}=-1, \\ \mathrm{co}\,\{ (1,1)\}, &\text{if}\ w_{1}>0,-1< w_{2}< 0, \\ \mathrm{co}\,\{ (1,1)\}, &\text{if}\ w_{1}>0, w_{2}\ge 0, \\ \mathrm{co}\,\{ (-1,1),(1,-1),(-1,-1)\}, &\text{if}\ w_{1}=0, w_{2}< -1, \\ \mathrm{co}\,\{ (1,1),(1,-1),(-1,1),(-1,-1)\}, &\text{if}\ w_{1}=0, w_{2}=-1, \\ \mathrm{co}\,\{ (-1,1),(1,1)\}, &\text{if}\ w_{1}=0, -1< w_{2}< 0, \\ \mathrm{co}\,\{ (1,1),(-1,1)\}, &\text{if}\ w_{1}=0, w_{2}\ge 0, \\ \mathrm{co}\,\{ (-1,-1)\}, &\text{if}\ w_{1}< 0, w_{2}< -1, \\ \mathrm{co}\,\{ (-1,1),(-1,-1)\}, &\text{if}\ w_{1}< 0, w_{2}=-1, \\ \mathrm{co}\,\{ (-1,1)\}, &\text{if}\ w_{1}< 0, -1< w_{2}< 0, \\ \mathrm{co}\,\{ (-1,1)\}, &\text{if}\ w_{1}< 0, w_{2}\ge 0. \end{cases}\displaystyle \end{aligned}$$

From all the above, we conclude that the conditions on the multipliers can be imposed as follows: $\eta _{1}^{g}\geq 0$, $\ \mu _{1}^{H}=0$, $\eta _{1}^{G}= 0$, $\alpha ^{L} w_{1}=-\eta _{1}^{G}=0$, and $\alpha ^{U} w_{2}=\eta _{1}^{g}$.

Then, we get the conclusions

$$\begin{aligned} F(w)+\eta _{1}^{g}g(w,0)-\eta _{1}^{G}G_{1}(w)- \eta _{1}^{H}H_{1}(w) &= 0+ \frac{(\eta _{1}^{g})^{2}}{2\alpha ^{U}}- \frac{(\eta _{1}^{g})^{2}}{\alpha ^{U}}+0+0 \\ &=-\frac{(\eta _{1}^{g})^{2}}{\alpha ^{U}} \\ &\leq \alpha ^{L}\frac{z_{1}^{2}}{2}+\alpha ^{U} \frac{z_{2}^{2}}{2} \\ &=F(z), \quad \forall \ z\in M. \end{aligned}$$

Hence, Theorem 3.2 is verified.

Now, we present an example to show that the obtained results may fail to hold when the convexificator is not available.

Example 3.2

Consider the following $(\mathit{WDISOPEC})$ problem in $\mathbb{R}^{2}$:

$$\begin{aligned} &\min F(z)=\bigl[F^{L}(z_{1},z_{2}), F^{U}(z_{1},z_{2})\bigr] = \bigl[ \lvert z_{1} \rvert , \lvert z_{2} \rvert \bigr], \\ &\quad \text{subject to} \quad g( z,t) = \vert z_{1} \vert + \vert z_{2} \vert -(t+1) \leq 0, \quad \forall t \in [0,1] , \\ &\hphantom{\quad \text{subject to} \quad} G_{1}(z ) = z _{1} \geq 0, \\ &\hphantom{\quad \text{subject to} \quad} H_{1}(z ) = z _{2} \geq 0, \\ &\hphantom{\quad \text{subject to} \quad} G_{1}(z ) H_{1}(z ) = z_{1}z_{2}=0 \quad \text{at } \bar{z}=(0,0). \end{aligned}$$

The feasible region of the problem is

$$\begin{aligned} M=\bigl\{ (z_{1},z_{2}) \in \mathbb{R}^{2} \mid z_{1}\geq 0, z_{2}\geq 0, z_{1}z_{2}=0, \lvert z_{1} \rvert + \lvert z_{2} \rvert \leq 1 \bigr\} . \end{aligned}$$

The Wolfe-type dual problem $(\mathit{WDISOPEC}(\bar{z}))$ for the primal problem is

$$ \begin{aligned} & \max_{w,\alpha ,\eta} \ F(w) + \eta _{t}^{g}g(w,t) - \eta _{1}^{G}G_{1}(w)- \eta _{1}^{H}H_{1}(w), \\ &\quad \text{subject to} \ \\ &\qquad 0 \in \alpha ^{L}\mathrm{co}\, \partial ^{*}F^{L}(w)+ \alpha ^{U} \mathrm{co}\, \partial ^{*}F^{U}(w) + \eta _{t}^{g} \mathrm{co}\, \partial ^{*}g(w,t) \\ &\qquad \qquad{} - \eta _{1}^{G} \mathrm{co}\,\partial ^{*}(G_{1}) (w)-\eta _{1}^{H} \mathrm{co}\, \partial ^{*}(H_{1}) (w), \\ &\alpha ^{L}, \alpha ^{U} \in (0,1), \quad \alpha ^{L}+ \alpha ^{U}= 1, \qquad \eta _{T_{g}}^{g} \geq 0, \qquad \eta _{j}^{h}\geq 0, \qquad \mu _{j}^{h} \geq 0, \quad j=1,\dots ,p, \\ &\eta _{i}^{G}\geq 0,\qquad \eta _{i}^{H} \geq 0,\qquad \mu _{i}^{G}\geq 0,\qquad \mu _{i}^{H} \ \geq 0, \quad i= 1,\dots ,l, \\ &\eta _{\gamma}^{G} = \eta _{\alpha}^{H} = \mu _{\gamma}^{G} = \mu _{ \alpha}^{H} = 0 ,\quad \forall i \in \beta , \qquad \mu _{i}^{G} = 0, \qquad \mu _{i}^{H} = 0, \\ & \vert w_{1} \vert + \vert w_{2} \vert -(t+1) \leq 0, \quad t \in T_{g}=\emptyset \quad \bigl( \implies \ \eta _{t}^{g}=0 \ \forall t\bigr), \\ &\quad \text{where } \partial ^{*}G_{1}(w)=\bigl\{ (1,0) \bigr\} , \qquad \partial ^{*}H_{1}(w)= \bigl\{ (0,1)\bigr\} , \end{aligned} $$

(27)

and

$$\begin{aligned} \partial ^{*} F^{L}(w) = \textstyle\begin{cases} \{(1,0)\}, & \ w_{1}> 0,\ w_{2}\in \mathbb{R}, \\ \{(1,0),(-1,0)\}, & \ w_{1}= 0, \ w_{2}\in \mathbb{R}, \\ \{(-1,0)\} & \ w_{1}< 0, \ w_{2}\in \mathbb{R}, \end{cases}\displaystyle \\ \partial ^{*} F^{U}(w) = \textstyle\begin{cases} \{(0,1)\}, & \ w_{1}\in \mathbb{R},\ w_{2}>0, \\ \{(0,1),(0,-1)\}, & \ w_{1}\in \mathbb{R}, \ w_{2}=0, \\ \{(0,-1)\} & \ w_{1} \in \mathbb{R}, \ w_{2}< 0. \end{cases}\displaystyle \end{aligned}$$

If we use a set which is not a convexificator then we show that the weak duality theorem does not hold. For this, let $\partial ^{*}F^{L}(w)=(1,2)$, $\partial ^{*} F^{U}(w) =(2,3)$. Then,

$$\begin{aligned}& (0,0)\in \alpha ^{L}(1,2)+\alpha ^{U}(2,3)-\eta _{1}^{G}(1,0)-\eta _{1}^{H}(0,1), \\& \quad \implies \quad \eta _{1}^{G}=\alpha ^{L}+2 \alpha ^{U},\qquad \eta _{1}^{H}=2 \alpha ^{L}+3\alpha ^{U}, \quad \text{and} \quad w=(w_{1},w_{2}) \in \mathbb{R}^{2}, \\& \quad \text{where } w=(w_{1},w_{2}) \quad \text{is arbitrary}. \end{aligned}$$

Now, we conclude that

$$ \begin{aligned} F(w)+\eta _{t}^{g}g(w,t)-\eta _{1}^{G}G_{1}(w)-\eta _{1}^{H}H_{1}(w) &= \alpha ^{L}\lvert w_{1} \rvert + \alpha ^{U}\lvert w_{2} \rvert - \eta _{1}^{G}w_{1}- \eta _{1}^{H}w_{2} \\ &\nleq \alpha ^{L}\lvert z_{1} \rvert +\alpha ^{U}\lvert z_{2} \rvert =F(z), \quad \forall \ z\in M, \ \end{aligned} $$

and w from the Wolfe dual feasible region.

As a direct consequence of Theorem 3.2, the following corollary follows.

Corollary 3.3

Let z̄ be feasible for (ISOPEC), where all constraint functions $g(\cdot ,t),\, t\in T;\ \pm h_{j}$, $j=1,\dots ,p;\, -G_{i}$, $i \in \nu _{1};\,-H_{i}$, $i\in \nu _{2}$ are affine and the index set $T_{g}$, as well as α, β, γ, is defined accordingly. Then, for any z feasible for (ISOPEC) and $(w,\alpha ^{L},\alpha ^{U}, \eta )$ feasible for (WDISOPEC), we have

$$ F(z) \nless ^{s}_{\mathrm{LU}} F(w)+\sum _{i=1}^{k}\eta _{i}^{g}g(w,t_{i}) +\sum_{j=1}^{p}\bigl(\eta _{j}^{h}-\mu _{j}^{h} \bigr)h_{j}(w)-\sum_{i=1}^{l} \bigl(\eta _{i}^{G}G_{i}(w)+ \eta _{i}^{H}H_{i}(w)\bigr). $$

Now, to obtain the strong duality result for the (WDISOPEC), we give the following corollary.

Corollary 3.4

Let z̄ be a local solution of (ISOPEC). Suppose that $F^{L}$, $F^{U}$ are locally Lipschitz around z̄ also that $F^{L}$, $F^{U}$ and effective constraint functions admitting bounded upper semiregular convexificators at z̄. If GS-ACQ is satisfied at z̄, then z̄ is a GS-stationary point.

Proof

If $F^{L}$, $F^{U}$, and effective constraint functions admit bounded upper semiregular convexificators, then, with the help of [9, Sect. 4.5] and [7, Theorem 3.1], z̄ is a GS-stationary point by Definition 2.8. □

Theorem 3.5

(Strong Duality)

Let z̄ be a local LU-optimal solution of (ISOPEC) and suppose $F^{L}$ and $F^{U}$ are locally Lipschitz near z̄. Suppose that $F^{L}$, $F^{U}$; $g(\cdot ,t)$, $t\in T$; $\pm h_{j}$, $j=1,\dots ,p$; $-G_{i}$, $i\in \nu _{1}$; $-H_{i}$, $i\in \nu _{2}$ admit bounded upper semiregular convexificators and are $\partial ^{*}$-convex functions at z̄. If GS-ACQ is satisfied at z̄, then there exist $(\bar{\alpha}^{L}, \bar{\alpha}^{U}) \in \mathbb{R}$ and $\bar{\eta} = (\bar{\eta}^{g}, \bar{\eta}^{h}, \bar{\eta}^{G}, \bar{\eta}^{H})\in \mathbb{R}^{k+p+2l}$ such that $(\bar{z},\bar{\alpha}^{L}, \bar{\alpha}^{U},\bar{\eta})$ is an LU-optimal solution of (WDISOPEC) and respective objective values are equal.

Proof

Since z̄ is a locally LU-optimal solution of (ISOPEC) and the GS-ACQ condition holds at z̄, by Corollary 3.4, $\exists \ (\bar{\alpha}^{L}, \bar{\alpha}^{U})\in \mathbb{R}$, $\bar{\eta} = (\bar{\eta}^{g}, \bar{\eta}^{h}, \bar{\eta}^{G}, \bar{\eta}^{H})\in \mathbb{R}^{k+p+2l}$, $\bar{\mu} =(\bar{\mu}^{h}, \bar{\mu}^{G}, \bar{\mu}^{H})\in \mathbb{R}^{p+2l}$, and indices $t_{1},t_{2},\dots ,t_{k} \in T_{g}(\bar{z})$, $k\leq n+1$ such that the GS-stationarity conditions for (ISOPEC) hold, that is, there exist $\bar{\xi}_{1} \in \mathrm{co}\,\partial ^{*}F^{L}(w) + \bar{\xi}_{2} \in \mathrm{co}\,\partial ^{*}F^{U}(w)$, $\bar{\xi}_{i}^{g} \in \mathrm{co}\,\partial ^{*}g(w,t_{i})$, $\bar{\zeta}_{j}^{h_{1}} \in \mathrm{co}\,\partial ^{*}h_{j}(w)$, $\bar{\zeta}_{j}^{h_{2}}\in \mathrm{co}\,\partial ^{*}(-h_{j})(w)$, $\bar{\xi _{i}}^{G} \in \mathrm{co}\,\partial ^{*}(-G_{i})(w)$, and $\bar{\xi}_{i}^{H} \in \mathrm{co}\,\partial ^{*}(-H_{i})(w)$ such that, for each $\bar{\xi}_{\alpha}=\bar{\alpha}^{L}\bar{\xi}_{1} + \bar{\alpha}^{U} \bar{\xi}_{2}$, one obtains

$$ \begin{aligned}&\bar{\xi}_{\alpha} + \sum \limits _{i=1}^{k}\bar{\eta}_{i}^{g} \bar{\xi _{i}}^{g} + \sum\limits _{j=1}^{p}\bigl[\bar{\eta}_{j}^{h} \bar{\zeta _{j}}^{h_{1}}+\bar{\mu}_{j}^{h} \bar{\zeta}_{j}^{h_{2}}\bigr] + \sum \limits _{i=1}^{l}\bigl[\bar{\eta}_{i}^{G} \bar{\xi _{i}}^{G}+ \bar{\eta _{i}}^{H} \bar{\xi _{i}}^{H}\bigr] = 0, \\ &\alpha ^{L}, \alpha ^{U} \in [0,1],\quad \bar{ \alpha}^{L}+ \bar{\alpha}^{U}= 1, \qquad \bar{ \eta}_{I_{g}}^{g} \geq 0,\\& \bar{\eta}_{j}^{h} \geq 0, \qquad \bar{\mu}_{j}^{h}\geq 0,\quad j=1,\dots ,p, \\ &\bar{\eta}_{i}^{G}\geq 0,\qquad \bar{\eta}_{i}^{H} \geq 0,\qquad \bar{\mu}_{i}^{G} \geq 0,\qquad \bar{ \mu}_{i}^{H} \geq 0, \quad i= 1,\dots ,l, \\ &\bar{\eta}_{\gamma}^{G} = \bar{\eta}_{\alpha}^{H} = \bar{\mu}_{ \gamma}^{G} = \bar{\mu}_{\alpha}^{H} = 0 ,\quad \forall i \in \beta , \qquad \bar{\mu}_{i}^{G} = 0, \qquad \bar{\mu}_{i}^{H} = 0. \end{aligned} $$

(28)

By the definition of (WDISOPEC), we suppose that $( \bar{z},\bar{\alpha}^{L}, \bar{\alpha}^{U},\bar{\eta})$ is a feasible vector of that problem. By virtue of Theorem 3.2, we have

$$ F(z) \nless ^{s}_{\mathrm{LU}} F(w)+\sum _{i=1}^{k}\eta _{i}^{g}g(w,t_{i}) +\sum_{j=1}^{p}\bigl(\eta _{j}^{h}-\mu _{j}^{h} \bigr)h_{j}(w)-\sum_{i=1}^{l} \bigl(\eta _{i}^{G}G_{i}(w)+ \eta _{i}^{H}H_{i}(w)\bigr) $$

(29)

for any feasible solution $(z,\alpha ^{L},\alpha ^{U},\eta )$ for the Wolfe dual (WDISOPEC). Since $t_{i} \in T_{g}(\bar{z})$, $g(\bar{z}, t_{i}) = 0$, $h_{j}(\bar{z})=0$, $\ G_{i}(\bar{z})=0$, $\forall i \in \nu _{1}$, $H_{i}(\bar{z})=0$, $\forall i \in \nu _{2}$, we have

$$ \begin{aligned}[b] \bar{\alpha}^{L}F^{L}( \bar{z})+ \bar{\alpha}^{U}F^{U}( \bar{z}) ={}& \bar{ \alpha}^{L}F^{L}(\bar{z})+ \bar{\alpha}^{U}F^{U}( \bar{z}) + \sum\limits _{i=1}^{k}\bar{ \eta}_{i}^{g}g(\bar{z},t_{i}) \\ &{} +\sum\limits _{j=1}^{p}\bigl(\bar{ \eta}_{j}^{h}-\bar{\mu}_{j}^{h} \bigr)h_{j}( \bar{z})-\sum\limits _{i=1}^{l} \bigl(\bar{\eta}_{i}^{G}G_{i}(\bar{z})+ \bar{\eta}_{i}^{H}H_{i}(\bar{z})\bigr). \end{aligned} $$

(30)

Now, from (29) and (30), we have

$$ \begin{aligned} &\bar{\alpha}^{L}F^{L}( \bar{z})+ \bar{\alpha}^{U}F^{U}( \bar{z}) + \sum \limits _{i=1}^{k}\bar{\eta}_{i}^{g}g( \bar{z},t_{i}) \\ &\qquad {} +\sum\limits _{j=1}^{p}\bigl(\bar{ \eta}_{j}^{h}-\bar{\mu}_{j}^{h} \bigr)h_{j}( \bar{z}) -\sum\limits _{i=1}^{l} \bigl(\bar{\eta}_{i}^{G}G_{i}(\bar{z})+ \bar{\eta}_{i}^{H}H_{i}(\bar{z})\bigr) \\ &\quad \nless ^{s}_{\mathrm{LU}} \ \alpha ^{L}F^{L}(w)+ \alpha ^{U}F^{U}(w)+ \sum\limits _{i=1}^{k} \eta _{i}^{g} g(w,t_{i}) \\ &\qquad {} +\sum\limits _{j=1}^{p}\bigl(\eta _{j}^{h}-\mu _{j}^{h} \bigr)h_{j}(w) -\sum\limits _{i=1} ^{l} \bigl(\eta _{i}^{G}G_{i}(w)+ \eta _{i}^{H}H_{i}(w)\bigr). \end{aligned} $$

Thus, the vector $(\bar{z},\bar{\alpha}^{L}, \bar{\alpha}^{U}),\bar{\eta})$ is an optimal solution for (WDISOPEC) and the respective objective values are equal, which completes the proof. □

By using convexificators, we establish duality results that relate the Mond–Weir-type dual problem (MWISOPEC) to the optimization problem with equilibrium constraints (ISOPEC).

Definition 3.6

(Mond–Weir dual)

The Mond–Weir dual (MWDISOPEC) for the interval -valued semiinfinite optimization problem with equilibrium constraints (ISOPEC) is defined by

$$\begin{aligned} &\max_{w,\alpha ,\eta} \ F(w)= \bigl[F^{L}(w),F^{U}(w) \bigr] \\ &\quad \text{subject to} \quad 0\in \alpha ^{L} \mathrm{co}\,\partial ^{*}F^{L}(w) + \alpha ^{U}\mathrm{co}\, \partial ^{*}F^{U}(w) + \sum_{i=1}^{k} \eta _{i}^{g}\mathrm{co}\,\partial ^{*}g(w,t_{i}) \\ &\hphantom{\quad \text{subject to} \quad 0\in}{} + \sum_{j=1}^{p}\bigl[\eta _{j}^{h}\mathrm{co}\,\partial ^{*}h_{j}(w)+ \mu _{j}^{h}\mathrm{co}\,\partial ^{*}(-h_{j}) (w)\bigr] \\ &\hphantom{\quad \text{subject to} \quad 0\in}{} + \sum_{i=1}^{l}\bigl[\eta _{i}^{G}\mathrm{co}\,\partial ^{*}(-G)_{i}(w)+ \eta _{i}^{H}\mathrm{co}\,\partial ^{*}(-H_{i}) (w)\bigr], \\ &g(w,t_{i}) \geq 0 \quad \bigl(t_{i} \in T_{g}(\bar{z})\bigr), \qquad h_{i}(w) = 0, \quad i=1,\dots ,p, \\ &G_{i}(w) \leq 0,\quad i \in \nu _{1},\qquad H_{i}(w) \leq 0,\quad i \in \nu _{2}, \\ &\alpha ^{L}, \alpha ^{U}\in [0,1],\quad \alpha ^{L}+ \alpha ^{U}= 1,\qquad \eta _{I_{g}}^{g} \geq 0,\qquad \eta _{j}^{h}\geq 0, \qquad \mu _{j}^{h}\geq 0, \quad j=1,\dots ,p, \\ &\eta _{i}^{G}\geq 0,\qquad \eta _{i}^{H} \geq 0,\qquad \mu _{i}^{G}\geq 0,\qquad \mu _{i}^{H} \geq 0, \quad i= 1,\dots ,l, \\ &\eta _{\gamma}^{G}= \eta _{\alpha}^{H} = \mu _{\gamma}^{G} = \mu _{ \alpha}^{H} = 0 ,\quad \forall i \in \beta , \qquad \mu _{i}^{G} = 0, \qquad \mu _{i}^{H} = 0, \end{aligned}$$

(31)

where $\eta = (\eta ^{g}, \eta ^{h}, \eta ^{G}, \eta ^{H}) \in \mathbb{R}^{k+p+2l}$, $\ \mu \in (\mu ^{h},\mu ^{G},\mu ^{H}) \in \mathbb{R}^{p+2l}$, and $t_{1},t_{2},\dots ,t_{k} \in T_{g}(\bar{z})$, $k \leq n+1$.

Theorem 3.7

(Weak duality)

Let z̄ and $(w,\alpha ^{L},\alpha ^{U},\eta )$ be the feasible points for the interval-valued semiinfinite optimization problem (ISOPEC) and the dual problem (MWDISOPEC), respectively, and let the index set $T_{g}$, as well as α, β, γ, be defined accordingly. Suppose the functions $F^{L}$, $F^{U}$; $g(\cdot ,t)$, $t\in T$; $\pm h_{j}$, $j=1,\dots ,p$; $-G_{i}$, $i\in \nu _{1}$; $-H_{i}$, $i\in \nu _{2}$ admit bounded upper semiregular convexificators and are $\partial ^{*}$-convex functions at w. If $L_{\mu} = \phi$, then

$$\begin{aligned} F(z) \nless ^{s}_{\mathrm{LU}} F(w). \end{aligned}$$

(32)

Proof

The proof is similar to that of Theorem 3.2. □

As a direct consequence of Theorem (3.7), the following corollary follows.

Corollary 3.8

Suppose z̄ is feasible for (ISOPEC) where all constraint functions $g(\cdot ,t),\,t\in T;\,\pm h_{j}$, $j=1,\dots ,p;\, -G_{i}$, $i\in \nu _{1};\,-H_{i}$, $i\in \nu _{2}$ are affine and the index set $T_{g}$, as well as α, β, γ, is defined accordingly. Then, for any z feasible for (ISOPEC) and w, α, η feasible for (WMDISOPEC), we have

$$\begin{aligned} F(z) \nless ^{s}_{\mathrm{LU}} F(w). \end{aligned}$$

(33)

Theorem 3.9

(Strong duality)

Suppose z̄ is a local LU-optimal solution of (ISOPEC) and $F^{L}$ and $F^{U}$ are locally Lipschitz at z̄. Suppose that $F^{L}$, $F^{U}$; $g(\cdot ,t)$, $t\in T$; $\pm h_{j}$, $j=1,\dots ,p$; $-G_{i}$, $i\in \nu _{1}$; $-H_{i}$, $i\in \nu _{2}$ admit bounded upper semiregular convexificators and are $\partial ^{*}$-convex functions at z̄. If GS-ACQ holds at z̄, then there exist $(\bar{\alpha}^{L}, \bar{\alpha}^{U}) \in \mathbb{R}$ and $\bar{\eta} = (\bar{\eta}^{g}, \bar{\eta}^{h}, \bar{\eta}^{G}, \bar{\eta}^{H})\in \mathbb{R}^{k+p+2l}$ such that $(\bar{z},\bar{\alpha}^{L}, \bar{\alpha}^{U},\bar{\eta})$ is an LU-optimal solution of (WMDISOPEC) and respective objective values are equal.

Proof

The proof is similar to that of Theorem 3.5. □

Now, we derive weak and strong results for the (ISOPEC) and its Mond–Weir-type dual problem under generalized $\partial ^{*}$-convexity assumptions.

Theorem 3.10

(Weak duality)

Let z̄ and $(w,\alpha ^{L}, \alpha ^{U},\eta )$ be feasible points for the interval-valued semiinfinite optimization problem (ISOPEC) and the dual problem (MWDISOPEC), respectively, and the index set $T_{g}$, as well as α, β, γ, be defined accordingly. Suppose the functions $F^{L}$ and $F^{U}$ are $\partial ^{*}$-pseudoconvex at w, $g(\cdot ,t)$, $t\in T$; $\pm h_{j}$, $j=1,\dots ,p$; $-G_{i}$, $i\in \nu _{1}$; $-H_{i}$, $i\in \nu _{2}$ admit bounded upper semiregular convexificators and are $\partial ^{*}$-quasiconvex functions at w. If $L_{\mu} = \phi$, then

$$\begin{aligned} F(z) \nless ^{s}_{\mathrm{LU}} F(w). \end{aligned}$$

(34)

Proof

Suppose that $F(z) <^{s}_{\mathrm{LU}} F(w)$, then we have

$$\begin{aligned} \bigl[F^{L}(z),F^{U}(z) \bigr]< ^{s}_{\mathrm{LU}} \bigl[F^{L}(w),F^{U}(w) \bigr]. \end{aligned}$$

(35)

For each $t_{i}\in T_{g}$,

$$\begin{aligned} g(z,t_{i}) \leq 0 \leq g(w,t_{i}), \end{aligned}$$

(36)

similarly,

$$\begin{aligned} &h_{j}(z)\leq h_{j}(w), \quad j=1,2,\dots ,p, \end{aligned}$$

(37)

$$\begin{aligned} &-h_{j}(z)\leq -h_{j}(w), \quad j=1,2,\dots ,p, \end{aligned}$$

(38)

$$\begin{aligned} &-G_{i}(z)\leq -G_{i}(w), \quad \forall \ i\in \nu _{1}, \end{aligned}$$

(39)

$$\begin{aligned} &-H_{i}(z)\leq -H_{i}(w), \quad \forall \ i\in \nu _{2} . \end{aligned}$$

(40)

Combining (36)–(40) and using $\partial ^{*}$-quasiconvexity of the above functions, we have

$$\begin{aligned} &\bigl\langle \xi _{i}^{g}, z-w \bigr\rangle \leq 0, \quad \forall \xi _{i}^{g} \in \partial ^{*}g(w,t_{i}), t_{i} \in T_{g}, \end{aligned}$$

(41)

$$\begin{aligned} &\bigl\langle \zeta _{j}^{h_{1}},z-w \bigr\rangle \leq 0, \quad \forall \zeta _{j}^{h_{1}} \in \partial ^{*}h_{j}(w), \forall j=1,\dots ,p, \end{aligned}$$

(42)

$$\begin{aligned} &\bigl\langle \zeta _{j}^{h_{2}},z-w \bigr\rangle \leq 0, \quad \forall \zeta _{j}^{h_{2}} \in \partial ^{*}(-h_{j}) (w), \forall j={1,\dots ,p}, \end{aligned}$$

(43)

$$\begin{aligned} &\bigl\langle \xi _{i}^{G},z-w \bigr\rangle \leq 0, \quad \forall \xi _{i}^{G} \in \partial ^{*}(-G_{i}) (w), \forall i\in \nu _{1}, \end{aligned}$$

(44)

$$\begin{aligned} &\bigl\langle \xi _{i}^{H},z-w \bigr\rangle \leq 0, \quad \forall \xi _{i}^{H} \in \partial ^{*}(-H_{i}) (w), \forall i\in \nu _{2}, \end{aligned}$$

(45)

that is,

$$\begin{aligned} \Biggl\langle \sum_{i=1}^{k}\eta _{i}^{g}\xi _{i}^{g} + \sum _{j=1}^{p}\bigl[\eta _{j}^{h}\zeta _{j}^{h_{1}}+\mu _{j}^{h}\zeta _{j}^{h_{2}}\bigr] + \sum_{i=1}^{l}\bigl[\eta _{i}^{G}\xi _{i}^{G}+\eta _{i}^{H}\xi _{i}^{H}\bigr], z-w \Biggr\rangle \leq 0. \end{aligned}$$

(46)

Using the above inequality and (7), there exist $\xi _{1}\in \partial ^{*}F^{L}(w)$ and $\xi _{2}\in \partial ^{*}F^{U}(w)$ such that

$$\begin{aligned} \bigl\langle \alpha ^{L}\xi _{1}+ \alpha ^{U}\xi _{2},z-w \bigr\rangle \geq 0. \end{aligned}$$

(47)

By (35) and the $\partial ^{*}$-pseudoconvexity of $F^{L}$ and $F^{U}$, it follows that

$$\begin{aligned} &\langle \xi _{1}, z-w\rangle < 0,\quad \forall \xi _{1} \in \partial ^{*}F^{L}(w), \\ &\langle \xi _{2}, z-w\rangle < 0,\quad \forall \xi _{2} \in \partial ^{*}F^{U}(w). \end{aligned}$$

Then $\langle \alpha ^{L}\xi _{1}+\alpha ^{U}\xi _{2},z-w \rangle <0$ and $\alpha ^{L}, \alpha ^{U} \in [0,1]$, $\alpha ^{L}+\alpha ^{U}=1$, which contradicts (47). This completes the proof. □

Theorem 3.11

(Strong duality)

Suppose z̄ is a local LU-optimal solution of (ISOPEC) and $F^{L}$ and $F^{U}$ are locally Lipschitz at z̄. Suppose that $F^{L}$, $F^{U}$ are $\partial ^{*}$-pseudoconvex at z̄, $g(\cdot ,t)$, $t\in T$; $\pm h_{j}$, $j=1,\dots ,p$; $-G_{i}$, $i\in \nu _{1}$; $-H_{i}$, $i\in \nu _{2}$ admit bounded upper semiregular convexificators and are $\partial ^{*}$-quasiconvex functions at z̄. If GS-ACQ holds at z̄, then there exist $(\bar{\alpha}^{L}, \bar{\alpha}^{U}) \in \mathbb{R}$ and $\bar{\eta} = (\bar{\eta}^{g}, \bar{\eta}^{h}, \bar{\eta}^{G}, \bar{\eta}^{H})\in \mathbb{R}^{k+p+2l}$ such that $(\bar{z},\bar{\alpha}^{L}, \bar{\alpha}^{U},\bar{\eta})$ is an LU-optimal solution of (WMDISOPEC) and respective objective values are equal.

Proof

The proof follows that of Theorem 3.9 and uses Theorem 3.10. □

4 Conclusion

In this paper, we have introduced duality results of interval-valued semiinfinite optimization problems with equilibrium constraints (ISOPEC). We have formulated the Wolfe-type dual (WDISOPEC) and Mond–Weir-type dual (WMDISOPEC) by using a convexificators. We have established weak and strong duality results relating to (ISOPEC) and the corresponding two duals (WDISOPEC) and (WMDISOPEC), under the assumption of $\partial ^{*}$-convexity, $\partial ^{*}$-pseudoconvexity, and $\partial ^{*}$-quasiconvexity. We can extend results of our paper for interval-valued semiinfinite optimization problems under assumptions of generalized convexity motivated by Sun et al. [35, 36].