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

Abstract

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.

1 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 semi-infinite 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 interval-valued 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.

2 Preliminary

Let X be a finite-dimensional Euclidean space. The notation $\langle \cdot ,\cdot \rangle$ denotes the inner product in X. For a point $\bar{x}\in X$, $B(\bar{x};\delta ):=\{x\in X:\|x-\bar{x}\|<\delta \}$ denotes the open ball of radius δ around x̄. For a set $C\subset X$, spanC, coneC stand for the linear hull and convex cone of C, respectively. Let $C\neq \emptyset$, the contingent cone of set C at the point x is defined by

$$ T(C,x):=\{v\in X\mid \exists t_{n}\rightarrow 0,\exists v_{n} \rightarrow v, \forall n\in N, x+t_{n}v_{n} \in C \}. $$

Let $\mathcal{D}$ be the set of all closed intervals in R. For any $A=[a_{1}, a_{2}]\in \mathcal{D}$, $B=[b_{1}, b_{2}]\in \mathcal{D}$, one has (see Moore [16])

$$\begin{aligned}& A+B=[a_{1}+b_{1}, a_{2}+b_{2}], \qquad -B=[-b_{2}, -b_{1}], \\& A-B=[a_{1}-b_{2}, a_{2}-b_{1}], \qquad A+k=[a_{1}+k, a_{2}+k], \end{aligned}$$

where k is any real number. A partial ordering for intervals can be formulated as follows:

$$\begin{aligned}& A \leq _{\mathit{LU}}B\quad \Longleftrightarrow\quad a_{1}\leq b_{1},\quad a_{2}\leq b_{2}, \\& A< _{\mathit{LU}}B\quad \Longleftrightarrow\quad A \leq _{\mathit{LU}}B, \quad A\neq B, \\& A\nless_{\mathit{LU}}B\text{ is the negation of } A< _{\mathit{LU}}B, \\& A< _{\mathit{LU}}^{s}B\quad \Longleftrightarrow\quad a_{1}< b_{1},\quad a_{2}< b_{2}, \\& A\nless_{\mathit{LU}}^{s}B\text{ is the negation of } A< _{\mathit{LU}}^{s}B. \end{aligned}$$

Let $F:X\rightarrow \mathcal{D}$ be a mapping on X defined as

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

where $F^{L}$, $F^{U}$ are the locally Lipschitz functions on X with $F^{L}(x)\leq F^{U}(x)$.

Now, we consider the following semi-infinite interval-valued optimization problem with vanishing constraints(IOPVC):

$$\begin{aligned} \mathit{LU}\text{-}\min \quad& F(x), \\ \text{s.t.}\quad& g_{j}(x)\leq 0,\quad j\in J, \\ &h_{k}(x)=0,\quad k=1,\ldots ,n, \\ &H_{i}(x)\geq 0,\quad i=1,\ldots ,l, \\ &G_{i}(x)H_{i}(x)\leq 0,\quad i=1,\ldots ,l, \end{aligned}$$

where $g_{j}:X\rightarrow R\cup \{+\infty \}$, $h_{k}:X\rightarrow R$, $H_{i}:X \rightarrow R$, $G_{i}:X\rightarrow R$ are the locally Lipschitz functions on X and the index set J is arbitrary(possibly infinite). Let $I_{n}:=\{1,\ldots ,n\}$, $I_{l}:=\{1,\ldots ,l\}$. The feasible set of problem (IOPVC) is

$$ E:=\bigl\{ x\in X:g_{j}(x)\leq 0(j\in J), h_{k}(x)=0(k \in I_{n}), H_{i}(x) \geq 0, G_{i}(x)H_{i}(x) \leq 0(i\in I_{l})\bigr\} . $$

Let $R_{+}^{|J|}$ denote the collection of all the functions $\lambda :J\rightarrow R$ taking values $\lambda _{j}>0$ only at finitely many points of J and equal to zero at the other points. For any $\bar{x}\in E$, $I_{g}(\bar{x}):=\{j\in J:g_{j}(\bar{x})=0 \}$ signifies the index set of all active constraints at x̄, and $\kappa (\bar{x}):=\{\lambda _{j}\in R_{+}^{|J|}:\lambda _{j}g_{j}( \bar{x})=0, \forall j\in J\}$ signifies the active constraint multipliers at x̄.

We give the following definitions of optimal solutions of (IOPVC).

Definition 2.1

([9])

Let $\bar{x}\in E$,

• (i) x̄ is said to be a locally LU optimal solution of (IOPVC) if there exists an open ball $B(\bar{x};\delta )$ such that there is no $x\in E\cap B(\bar{x};\delta )$ satisfying

  $$ F(x)< _{\mathit{LU}}F(\bar{x}). $$

• (ii) x̄ is said to be a locally weakly LU optimal solution of (IOPVC) if there exists an open ball $B(\bar{x};\delta )$ such that there is no $x\in E\cap B(\bar{x};\delta )$ satisfying

  $$ F(x)< _{\mathit{LU}}^{s}F(\bar{x}). $$

Let $f:X\rightarrow R$ be a locally Lipschitz function around x̄. The Clarke directional derivative of f around x̄ in the direction $v\in X$ and the Clarke subdifferential of f at x̄ are, respectively, given by (see Clarke [17])

$$\begin{aligned}& f_{c}^{\prime }(\bar{x};v):=\limsup_{x\rightarrow \bar{x},t \downarrow 0} \frac{f(x+\mathit{tv})-f(x)}{t}, \\& \partial _{c}f(\bar{x}):=\bigl\{ \xi \in X:\langle \xi ,v\rangle \leq f_{c}^{\prime }( \bar{x};v), \forall v\in X\bigr\} . \end{aligned}$$

Definition 2.2

([3])

Let $f:X\rightarrow R$ be a locally Lipschitz function around x̄. Then

• (i) f is said to be $\partial _{c}$-pseudoconvex at x̄ if, for each $x\in X$ and any $\xi \in \partial _{c}f(\bar{x})$,

  $$ f(x)-f(\bar{x})< 0\quad \Rightarrow \quad \langle \xi , x-\bar{x}\rangle < 0; $$

• (ii) f is said to be strictly $\partial _{c}$-pseudoconvex at x̄ if, for each $x\in X$, $x\neq \bar{x}$ and any $\xi \in \partial _{c}f(\bar{x})$,

  $$ f(x)-f(\bar{x})\leq 0\quad \Rightarrow\quad \langle \xi , x-\bar{x}\rangle < 0; $$

• (iii) f is said to be $\partial _{c}$-quasiconvex at x̄ if, for each $x\in X$ and any $\xi \in \partial _{c}f(\bar{x})$,

  $$ f(x)-f(\bar{x})\leq 0\quad \Rightarrow \quad \langle \xi , x-\bar{x}\rangle \leq 0. $$

The following sets of indicators, which will be used in the sequel, are given. Let $x\in E$,

$$\begin{aligned}& I_{+}(x):=\bigl\{ i\in I_{l}\mid H_{i}(x)>0 \bigr\} ,\qquad I_{0}(x):=\bigl\{ i\in I_{l} \mid H_{i}(x)=0\bigr\} , \\& I_{+0}(x):=\bigl\{ i\in I_{l}\mid H_{i}(x)>0,G_{i}(x)=0 \bigr\} , \\& I_{+-}(x):=\bigl\{ i\in I_{l}\mid H_{i}(x)>0,G_{i}(x)< 0 \bigr\} , \\& I_{0+}(x):=\bigl\{ i\in I_{l}\mid H_{i}(x)=0,G_{i}(x)>0 \bigr\} , \\& I_{00}(x):=\bigl\{ i\in I_{l}\mid H_{i}(x)=0,G_{i}(x)=0 \bigr\} , \\& I_{0-}(x):=\bigl\{ i\in I_{l}\mid H_{i}(x)=0,G_{i}(x)< 0 \bigr\} . \end{aligned}$$

Referring to Definition 4 in [4], we give the following definition.

Definition 2.3

Let $\bar{x}\in E$ be a feasible point of (IOPVC).

• (i) The Abadie constraint qualification (ACQ) is said to hold at x̄ iff $T(E, \bar{x})=L(\bar{x})$, where $L(\bar{x})$ is the linearized cone of (IOPVC) at x̄, and

  $$\begin{aligned} L(\bar{x}):={}&\bigl\{ v\in X\mid \bigl\langle \xi _{j}^{g},v\bigr\rangle \leq 0, \forall \xi _{j}^{g}\in \partial _{c}g_{j}( \bar{x}), j\in I_{g}( \bar{x}); \bigl\langle \xi _{k}^{h},v \bigr\rangle =0, \forall \xi _{k}^{h} \in \partial _{c}h_{k}(\bar{x}), \\ &{}k\in I_{n}; \bigl\langle \xi _{i}^{H},v \bigr\rangle =0, \forall \xi _{i}^{H}\in \partial _{c}H_{i}(\bar{x}), i\in I_{0+}; \bigl\langle \xi _{i}^{H},v\bigr\rangle \geq 0, \forall \xi _{i}^{H}\in \partial _{c}H_{i}( \bar{x}), \\ &{}i\in I_{00}\cup I_{0-}; \bigl\langle \xi _{i}^{G},v\bigr\rangle \leq 0, \forall \xi _{i}^{G}\in \partial _{c}G_{i}( \bar{x}), i\in I_{+0} \bigr\} . \end{aligned}$$

  (2.1)

• (ii) The VC-ACQ is said to hold at x̄ iff $L_{\mathit{VC}}(\bar{x})\subseteq T(E, \bar{x})$, where $L_{\mathit{VC}}(\bar{x})$ is the corresponding VC-linearized cone of (IOPVC) at x̄, and

  $$ \begin{aligned}[b] L_{\mathit{VC}}(\bar{x}):={}& \bigl\{ v\in X\mid \bigl\langle \xi _{j}^{g},v\bigr\rangle \leq 0, \forall \xi _{j}^{g}\in \partial _{c}g_{j}(\bar{x}), j\in I_{g}( \bar{x}); \\ & \bigl\langle \xi _{k}^{h},v\bigr\rangle =0, \forall \xi _{k}^{h}\in \partial _{c}h_{k}( \bar{x}), k\in I_{n}; \\ & \bigl\langle \xi _{i}^{H},v\bigr\rangle =0, \forall \xi _{i}^{H}\in \partial _{c}H_{i}( \bar{x}), i\in I_{0+}; \\ & \bigl\langle \xi _{i}^{H},v\bigr\rangle \geq 0, \forall \xi _{i}^{H}\in \partial _{c}H_{i}( \bar{x}), i\in I_{00}\cup I_{0-}; \\ & \bigl\langle \xi _{i}^{G},v\bigr\rangle \leq 0, \forall \xi _{i}^{G}\in \partial _{c}G_{i}( \bar{x}), i\in I_{+0}\cup I_{00} \bigr\} .\end{aligned} $$

  (2.2)

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

Let $\bar{x}\in E$ be a locally weakly LU optimal solution of (IOPVC) such that (VC-ACQ) holds at x̄ and

$$\begin{aligned} \triangle :={}&\operatorname{cone}\biggl(\bigcup_{j\in I_{g}(\bar{x})} \partial _{c}g_{j}( \bar{x})\cup \bigcup _{i\in I_{00}\cup I_{0-}}-\partial _{c}H_{i}( \bar{x}) \cup \bigcup_{i\in I_{+0}\cup I_{00}}\partial _{c}G_{i}( \bar{x})\biggr) \\ &{}+\operatorname{span}\biggl(\bigcup_{k\in I_{n}} \partial _{c}h_{k}(\bar{x})\cup \bigcup _{i\in I_{0+}}\partial _{c}H_{i}(\bar{x}) \biggr) \end{aligned}$$

is closed. Then there exist Lagrange multipliers $\alpha ^{L}, \alpha ^{U}\in R_{+}$, $\lambda ^{g}\in \kappa (\bar{x})$, $\lambda ^{h} \in R^{n}$, $\lambda ^{H}, \lambda ^{G}\in R^{l}$ such that

$$ \begin{aligned}[b] 0\in {}&\alpha ^{L} \partial _{c}F^{L}(\bar{x})+\alpha ^{U} \partial _{c}F^{U}( \bar{x}) +\sum _{j\in J}\lambda _{j}^{g}\partial _{c}g_{j}( \bar{x})+\sum_{k=1}^{n} \lambda _{k}^{h}\partial _{c}h_{k}( \bar{x}) \\ &{}-\sum_{i=1}^{l}\lambda _{i}^{H}\partial _{c}H_{i}( \bar{x}) + \sum_{i=1}^{l}\lambda _{i}^{G}\partial _{c}G_{i}( \bar{x}) \end{aligned} $$

(2.3)

and

$$ \textstyle\begin{cases} \alpha ^{L}+\alpha ^{U}=1,\qquad h_{k}(\bar{x})=0\quad (k\in I_{n}), \\ \lambda _{j}^{g}\geq 0,\quad g_{j}(\bar{x})\leq 0,\qquad \lambda _{j}^{g}g_{j}( \bar{x})=0\quad (j\in J), \\ \lambda _{i}^{H}=0\quad (i\in I_{+}(\bar{x})),\qquad \lambda _{i}^{H}\geq 0\quad (i \in I_{00}(\bar{x})\cup I_{0-}(\bar{x})),\\ \lambda _{i}^{H}\in R\quad (i \in I_{0+}(\bar{x})), \\ \lambda _{i}^{G}=0\quad (i\in I_{0+}(\bar{x})\cup I_{0-}(\bar{x})\cup I_{+-}( \bar{x})), \qquad \lambda _{i}^{G}\geq 0\quad (i\in I_{00}(\bar{x})\cup I_{+0}( \bar{x})). \end{cases} $$

(2.4)

Definition 2.4

([4])

The point x is said to be a VC-stationary point of (IOPVC) if there exist Lagrange multipliers $\alpha ^{L}, \alpha ^{U}\in R_{+}$, $\lambda ^{g}\in \kappa (x)$, $\lambda ^{h}\in R^{n}$, $\lambda ^{H}, \lambda ^{G}\in R^{l}$ such that (2.3) and (2.4) hold.

Now, let x be a VC-stationary point of (IOPVC) with corresponding multipliers $\lambda ^{g}\in R_{+}^{|J|}$, $\lambda ^{h}\in R^{n}$, $\lambda ^{H}, \lambda ^{G}\in R^{l}$, we give the following index sets:

$$\begin{aligned}& I_{g}^{+}(x):=\bigl\{ j\in I_{g}(x)\mid \lambda _{j}^{g}>0 \bigr\} , \\& I_{h}^{+}(x):=\bigl\{ k\in I_{n}(x)\mid \lambda _{k}^{h}>0 \bigr\} ,\qquad I_{h}^{-}(x):= \bigl\{ k\in I_{n}(x)\mid \lambda _{k}^{h}< 0 \bigr\} , \\& I_{00}^{+}(x):=\bigl\{ i\in I_{00}(x)\mid \lambda _{i}^{H}>0 \bigr\} ,\qquad I_{0-}^{+}(x):= \bigl\{ i\in I_{0-}(x)\mid \lambda _{i}^{H}>0 \bigr\} , \\& I_{0+}^{+}(x):=\bigl\{ i\in I_{0+}(x)\mid \lambda _{i}^{H}>0 \bigr\} ,\qquad I_{0+}^{-}(x):= \bigl\{ i\in I_{0+}(x)\mid \lambda _{i}^{H}< 0 \bigr\} , \\& I_{+}^{+}(x):=\bigl\{ i\in I_{+}(x)\mid \lambda _{i}^{H}>0 \bigr\} ,\qquad \tilde{I}_{+}^{+}(x):= \bigl\{ i\in I_{+}(x)\mid \lambda _{i}^{G}>0 \bigr\} . \end{aligned}$$

3 Wolfe type duality

In this section, we refer to [6] to give the following Wolfe type dual models. First of all, let $\lambda ^{g}\in R_{+}^{|J|}$, $\lambda ^{h}\in R^{n}$, $\lambda ^{H}, \lambda ^{G}\in R^{l}$,

$$\begin{aligned} \Phi \bigl(\cdot , \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr)={}&F(\cdot )+ \sum_{j\in J} \lambda _{j}^{g}g_{j}(\cdot )+\sum _{k=1}^{n}\lambda _{k}^{h}h_{k}( \cdot ) \\ &{}-\sum_{i=1}^{l}\lambda _{i}^{H}H_{i}(\cdot )+\sum _{i=1}^{l} \lambda _{i}^{G}G_{i}( \cdot ) \end{aligned}$$

is an interval-valued function, and

$$\begin{aligned} \Delta (\cdot ):={}&\alpha ^{L}\partial _{c}F^{L}( \cdot )+\alpha ^{U} \partial _{c}F^{U}(\cdot ) +\sum_{j\in J}\lambda _{j}^{g} \partial _{c}g_{j}(\cdot )+\sum _{k=1}^{n} \lambda _{k}^{h} \partial _{c}h_{k}(\cdot ) \\ &{}-\sum_{i=1}^{l}\lambda _{i}^{H}\partial _{c}H_{i}( \cdot ) + \sum_{i=1}^{l}\lambda _{i}^{G}\partial _{c}G_{i}( \cdot ). \end{aligned}$$

Now, we give the Wolfe type dual model of (IOPVC). For $x\in E$,

$$\begin{aligned}& \bigl(D_{W}(x)\bigr)\quad \mathit{LU}\text{-}\max \quad \Phi \bigl(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr) \\& \quad \textstyle\begin{cases} s.t.\quad 0\in \Delta (y) \\ \alpha ^{L}, \alpha ^{U}\in R_{+}, \quad \alpha ^{L}+\alpha ^{U}=1, \\ \lambda _{j}^{g}\geq 0, \quad \forall j\in J, \\ \lambda _{i}^{G}=v_{i}H_{i}(x), v_{i}\geq 0, \quad \forall i\in I_{l}, \\ \lambda _{i}^{H}=\varrho _{i}-v_{i}G_{i}(x),\quad \varrho _{i}\geq 0, \forall i\in I_{l}. \end{cases}\displaystyle \\& \begin{aligned} E_{W}(x):={}&\bigl\{ \bigl(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h},\lambda ^{H}, \lambda ^{G}, \varrho , v \bigr): \\ &0\in \Delta (y), y\in X, \alpha ^{L}, \alpha ^{U}\in R_{+}, \alpha ^{L}+\alpha ^{U}=1, \\ &\lambda _{j}^{g}\geq 0, \forall j\in J, \lambda _{i}^{G}=v_{i}H_{i}(x), v_{i}\geq 0, \forall i\in I_{l} \\ &\lambda _{i}^{H}=\varrho _{i}-v_{i}G_{i}(x), \varrho _{i}\geq 0, \forall i\in I_{l}\bigr\} \end{aligned} \end{aligned}$$

(3.1)

denotes the feasible set of $(D_{W}(x))$. $\mathit{prE}_{W}(x):=\{y\in X:(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{W}(x)\}$ represents the projection of the set $E_{W}(x)$ on X.

In order to be independent of (IOPVC), we give another Wolfe type dual model:

$$\begin{aligned}& (D_{W}) \mathit{LU}\text{-}\max \quad \Phi \bigl(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr) \\& \quad \text{s.t. } \bigl(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v\bigr)\in E_{W}:=\bigcap_{x\in E}E_{W}(x), \end{aligned}$$

where $E_{W}$ denotes the set of all feasible points of $(D_{W})$ and $\mathit{prE}_{W}$ denotes the projection of the set $E_{W}$ on X.

Definition 3.1

([4])

Let $x\in E$.

• (i) $(\bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{W}(x)$ is said to be a locally LU optimal solution of $(D_{W}(x))$ if there exists $B(\bar{y};\delta )$ such that there is no $y\in E_{W}(x)\cap B(\bar{y};\delta )$ satisfying

  $$ \Phi \bigl(\bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr)< _{\mathit{LU}}\Phi \bigl(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr). $$

• (ii) $(\bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{W}(x)$ is said to be a locally weakly LU optimal solution of $(D_{W}(x))$ if there exists $B(\bar{y};\delta )$ such that there is no $y\in E_{W}(x)\cap B(\bar{y};\delta )$ satisfying

  $$ \Phi \bigl(\bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr)< _{\mathit{LU}}^{s}\Phi \bigl(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr). $$

Theorem 3.1

(Weak duality)

Let $x\in E$, $(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{W}$ be feasible points for the (IOPVC) and the $(D_{W})$, respectively. If $\Phi ^{L}(\cdot , \alpha ^{L}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$, $\Phi ^{U}(\cdot , \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$ are $\partial _{c}$-pseudoconvex at $y\in E\cup \mathit{prE}_{W}$, then

$$ F(x)\nless_{\mathit{LU}}^{s}\Phi \bigl(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr). $$

Proof

Suppose $F(x)<_{\mathit{LU}}^{s}\Phi (y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$, then

$$ F(x)< _{\mathit{LU}}^{s}F(y)+\sum_{j\in J} \lambda _{j}^{g}g_{j}(y)+ \sum _{k=1}^{n}\lambda _{k}^{h}h_{k}(y) -\sum_{i=1}^{l} \lambda _{i}^{H}H_{i}(y)+\sum _{i=1}^{l}\lambda _{i}^{G}G_{i}(y). $$

(3.2)

Since $x\in E$ and $(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{W}$, it follows that

$$ \begin{gathered} g_{j}(x)< 0,\qquad \lambda _{j}^{g}\geq 0,\quad j \notin I_{g}(x), \\ g_{j}(x)=0,\qquad \lambda _{j}^{g}\geq 0, \quad j\in I_{g}(x), \\ h_{k}(x)=0,\qquad \lambda _{k}^{h}\in R, \quad k\in I_{n}, \\ -H_{i}(x)< 0,\qquad \lambda _{i}^{H}\geq 0, \quad i\in I_{+}(x), \\ -H_{i}(x)=0,\qquad \lambda _{i}^{H}\in R, \quad i\in I_{0}(x), \\ G_{i}(x)>0,\qquad \lambda _{i}^{G}=0,\quad i\in I_{0+}(x), \\ G_{i}(x)=0,\qquad \lambda _{i}^{G}\geq 0, \quad i\in I_{00}(x) \cup I_{+0}(x), \\ G_{i}(x)< 0,\qquad \lambda _{i}^{G}\geq 0, \quad i\in I_{0-}(x) \cup I_{+-}(x). \end{gathered} $$

(3.3)

The above formulas imply that

$$ \sum_{j\in J}\lambda _{j}^{g}g_{j}(x)+ \sum_{k=1}^{n} \lambda _{k}^{h}h_{k}(x)- \sum _{i=1}^{l}\lambda _{i}^{H}H_{i}(x)+ \sum_{i=1}^{l}\lambda _{i}^{G}G_{i}(x) \leq 0. $$

(3.4)

Equation (3.4) together with (3.2) proves that

$$ \Phi \bigl(x, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr)< _{\mathit{LU}}^{s}\Phi \bigl(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr). $$

By the $\partial _{c}$-pseudoconvexity of $\Phi ^{L}(\cdot , \alpha ^{L}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$, $\Phi ^{U}(\cdot , \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$ at $y\in E\cup \mathit{prE}_{W}$, it follows that

$$ \begin{gathered} \Biggl\langle \xi ^{L}+ \sum_{j\in J}\lambda _{j}^{g} \xi _{j}^{g}+ \sum_{k=1}^{n} \lambda _{k}^{h}\xi _{k}^{h}- \sum_{i=1}^{l} \lambda _{i}^{H}\xi _{i}^{H}+\sum _{i=1}^{l}\lambda _{i}^{G} \xi _{i}^{G}, x-y \Biggr\rangle < 0, \\ \Biggl\langle \xi ^{U}+\sum_{j\in J} \lambda _{j}^{g}\xi _{j}^{g}+ \sum_{k=1}^{n}\lambda _{k}^{h}\xi _{k}^{h}- \sum _{i=1}^{l} \lambda _{i}^{H} \xi _{i}^{H}+\sum_{i=1}^{l} \lambda _{i}^{G} \xi _{i}^{G}, x-y \Biggr\rangle < 0, \end{gathered} $$

(3.5)

where $\xi ^{L}\in \partial _{c}F^{L}(y)$, $\xi ^{U}\in \partial _{c}F^{U}(y)$, $\xi _{j}^{g}\in \partial _{c}g_{j}(y)$, $j\in J$, $\xi _{k}^{h}\in \partial _{c}h_{k}(y)$, $k\in I_{n}$, $\xi _{i}^{H}\in \partial _{c}H_{i}(y)$, $i\in I_{l}$, $\xi _{i}^{G}\in \partial _{c}G_{i}(y)$, $i\in I_{l}$. Add $\alpha ^{L}$ and $\alpha ^{U}$ to both sides of inequations (3.5)

$$ \langle \xi , x-y\rangle < 0, \quad \forall \xi \in \Delta (y), $$

contradicting $0\in \Delta (y)$, the result is proved. □

Theorem 3.2

(Weak duality)

Let $x\in E$, $(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{W}$ be feasible points for the (IOPVC) and the $(D_{W})$, respectively. If $\Phi ^{L}(\cdot , \alpha ^{L}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$, $\Phi ^{U}(\cdot , \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$ are strictly $\partial _{c}$-pseudoconvex at $y\in E\cup \mathit{prE}_{W}$, then

$$ F(x)\nless_{\mathit{LU}}\Phi \bigl(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr). $$

Proof

The proof is similar to Theorem 3.1. □

Theorem 3.3

(Strong duality)

Let $\bar{x}\in E$ be a locally weakly LU optimal solution of (IOPVC) such that the (VC-ACQ) holds at x̄ and △ is closed. Then there exist Lagrange multipliers $\bar{\alpha }^{L}, \bar{\alpha }^{U}\in R_{+}$, $\bar{\lambda }^{g}\in R_{+}^{|J|}$, $\bar{\lambda }^{h}\in R^{n}$, $\bar{\lambda }^{H}, \bar{\lambda }^{G}, \bar{\varrho }, \bar{v}\in R^{l}$ such that $(\bar{x}, \bar{\alpha }^{L}, \bar{\alpha }^{U}, \bar{\lambda }^{g}, \bar{\lambda }^{h}, \bar{\lambda }^{H}, \bar{\lambda }^{G}, \bar{\varrho }, \bar{v})$ is a feasible point of $(D_{W}(\bar{x}))$, and

$$ F(\bar{x})=\Phi \bigl(\bar{x}, \bar{\alpha }^{L}, \bar{\alpha }^{U}, \bar{\lambda }^{g}, \bar{\lambda }^{h}, \bar{\lambda }^{H}, \bar{\lambda }^{G} \bigr). $$

Moreover, if $\Phi ^{L}(\cdot , \alpha ^{L}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$, $\Phi ^{U}(\cdot , \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$ are $\partial _{c}$-pseudoconvex at $y\in E\cup \mathit{prE}_{W}(\bar{x})$, then $(\bar{x}, \bar{\alpha }^{L}, \bar{\alpha }^{U}, \bar{\lambda }^{g}, \bar{\lambda }^{h}, \bar{\lambda }^{H}, \bar{\lambda }^{G}, \bar{\varrho }, \bar{v})$ is a locally weakly LU optimal solution of $(D_{W}(\bar{x}))$.

Proof

By Theorem 2.1, it follows that there exist Lagrange multipliers $\bar{\alpha }^{L}, \bar{\alpha }^{U}\in R_{+}$, $\bar{\lambda }^{g}\in \kappa (\bar{x})$, $\bar{\lambda }^{h}\in R^{n}$, $\bar{\lambda }^{H}, \bar{\lambda }^{G}, \bar{\varrho }, \bar{v}\in R^{l}$ such that (2.3) and (2.4) are satisfied. Combined with the definition of $(D_{W}(\bar{x}))$, one has that $(\bar{x}, \bar{\alpha }^{L}, \bar{\alpha }^{U}, \bar{\lambda }^{g}, \bar{\lambda }^{h}, \bar{\lambda }^{H}, \bar{\lambda }^{G}, \bar{\varrho }, \bar{v})$ is a feasible point of $(D_{W}(\bar{x}))$,

$$ \sum_{j\in J}\bar{\lambda }_{j}^{g}g_{j}( \bar{x})+\sum_{k=1}^{n}\bar{\lambda }_{k}^{h}h_{k} (\bar{x})-\sum _{i=1}^{l} \bar{\lambda }_{i}^{H}H_{i}( \bar{x})+\sum_{i=1}^{l} \bar{\lambda }_{i}^{G}G_{i}(\bar{x})=0 $$

and

$$ F(\bar{x})=\Phi \bigl(\bar{x}, \bar{\alpha }^{L}, \bar{\alpha }^{U}, \bar{\lambda }^{g}, \bar{\lambda }^{h}, \bar{\lambda }^{H}, \bar{\lambda }^{G} \bigr). $$

Then, from Theorem 3.1 one has, for any $(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{W}(\bar{x})$,

$$ \Phi \bigl(\bar{x}, \bar{\alpha }^{L}, \bar{\alpha }^{U}, \bar{\lambda }^{g}, \bar{\lambda }^{h}, \bar{\lambda }^{H}, \bar{\lambda }^{G} \bigr)=F(\bar{x}) \nless_{\mathit{LU}}^{s}\Phi \bigl(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr). $$

Therefore, $(\bar{x}, \bar{\alpha }^{L}, \bar{\alpha }^{U}, \bar{\lambda }^{g}, \bar{\lambda }^{h}, \bar{\lambda }^{H}, \bar{\lambda }^{G}, \bar{\varrho },\bar{v})$ is a locally weakly LU optimal solution of $(D_{W}(\bar{x}))$. □

Theorem 3.4

(Converse duality)

Let $x\in E$ be any feasible point of (IOPVC), and let $(\bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{W}$ be a feasible point of $(D_{W})$ such that

$$ \textstyle\begin{cases} \lambda _{j}^{g}g_{j}(\bar{y})\geq 0,&\forall j\in J, \\ \lambda _{k}^{h}h_{k}(\bar{y})=0,&\forall k\in I_{n}, \\ -\lambda _{i}^{H}H_{i}(\bar{y})\geq 0,&\forall i\in I_{l}, \\ \lambda _{i}^{G}G_{i}(\bar{y})\geq 0,&\forall i\in I_{m}. \end{cases} $$

(3.6)

If one of the following conditions holds:

• (i) $\Phi ^{L}(\cdot , \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$, $\Phi ^{U}(\cdot , \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$ are $\partial _{c}$-pseudoconvex at $\bar{y}\in E\cup \mathit{prE}_{W}$;

• (ii) $F^{L}(\cdot )$, $F^{U}(\cdot )$ are $\partial _{c}$-pseudoconvex at $\bar{y}\in E\cup \mathit{prE}_{W}(x)$, $g_{j}$ ($j\in I_{g}^{+}(x)$), $h_{k}$ ($k\in I_{h}^{+}(x)$), $-h_{k}$ ($k \in I_{h}^{-}(x)$), $-H_{i}$ ($i\in I_{+}^{+}(x)\cup I_{00}^{+}(x)\cup I_{0-}^{+}(x) \cup I_{0+}^{+}(x)$), $H_{i}$ ($i\in I_{0+}^{-}(x)$), $G_{i}$ ($i\in \tilde{I}_{+}^{+}(x)$) are $\partial _{c}$-quasiconvex at $\bar{y}\in E\cup \mathit{prE}_{W}$;

Then ȳ is the locally weakly LU optimal solution of (IOPVC).

Proof

Suppose to the contrary that ȳ is not a locally weakly LU optimal solution of (IOPVC), then there exists $\bar{x}\in E\cap B(\bar{y};\delta )$ such that

$$ F(\bar{x})< _{\mathit{LU}}^{s}F(\bar{y}). $$

(3.7)

(i) Since $\bar{x}\in E$ and $(\bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)$ are feasible points for the (IOPVC) and the $(D_{W})$, respectively. Therefore, combined with (3.3) and (3.6), one gets

$$\begin{aligned}& \sum_{j\in J}\lambda _{j}^{g}g_{j}( \bar{x})+\sum_{k=1}^{n} \lambda _{k}^{h}h_{k}(\bar{x})- \sum _{i=1}^{l}\lambda _{i}^{H}H_{i}( \bar{x})+\sum_{i=1}^{l}\lambda _{i}^{G}G_{i}(\bar{x}) \\& \quad \leq 0\leq \sum_{j\in J}\lambda _{j}^{g}g_{j}(\bar{y})+\sum _{k=1}^{n}\lambda _{k}^{h}h_{k}( \bar{y})- \sum_{i=1}^{l} \lambda _{i}^{H}H_{i}(\bar{y})+\sum _{i=1}^{l}\lambda _{i}^{G}G_{i}( \bar{y}). \end{aligned}$$

(3.8)

By (3.7) and (3.8), one has

$$ \Phi \bigl(\bar{x}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr)< _{\mathit{LU}}^{s}\Phi \bigl(\bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G} \bigr). $$

And by the $\partial _{c}$-pseudoconvexity of $\Phi ^{L}(\cdot , \alpha ^{L}, \alpha ^{U} \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$, $\Phi ^{U}(\cdot , \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$ at $\bar{y}\in E\cup \mathit{prE}_{W}$, one has

$$ \textstyle\begin{cases} \langle \xi ^{L}+\sum_{j\in J}\lambda _{j}^{g}\xi _{j}^{g}+ \sum_{k=1}^{n}\lambda _{k}^{h}\xi _{k}^{h}- \sum_{i=1}^{l} \lambda _{i}^{H}\xi _{i}^{H}+\sum_{i=1}^{l}\lambda _{i}^{G} \xi _{i}^{G}, \bar{x}-\bar{y} \rangle < 0, \\ \langle \xi ^{U}+\sum_{j\in J}\lambda _{j}^{g}\xi _{j}^{g}+ \sum_{k=1}^{n}\lambda _{k}^{h}\xi _{k}^{h}- \sum_{i=1}^{l} \lambda _{i}^{H}\xi _{i}^{H}+\sum_{i=1}^{l}\lambda _{i}^{G} \xi _{i}^{G}, \bar{x}-\bar{y} \rangle < 0, \end{cases} $$

(3.9)

where $\xi ^{L}\in \partial _{c}F^{L}(\bar{y})$, $\xi ^{U}\in \partial _{c}F^{U}(\bar{y})$, $\xi _{j}^{g}\in \partial _{c}g_{j}( \bar{y})$, $j\in J$, $\xi _{k}^{h}\in \partial _{c}h_{k}(\bar{y})$, $k \in I_{n}$, $\xi _{i}^{H}\in \partial _{c}H_{i}(\bar{y})$, $i\in I_{l}$, $\xi _{i}^{G}\in \partial _{c}G_{i}(\bar{y})$, $i\in I_{l}$. Combining (3.9) with $\alpha ^{L}$ and $\alpha ^{U}$, we get

$$ \langle \xi , \bar{x}-\bar{y}\rangle < 0,\quad \forall \xi \in \Delta (\bar{y}), $$

which contradicts $0\in \Delta (\bar{y})$, hence the result holds.

(ii) Since $\bar{x}\in E$ and $(\bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{W}$, by (3.3) and (3.6), one has

$$\begin{aligned}& \lambda _{j}^{g}g_{j}(\bar{x})\leq \lambda _{j}^{g}g_{j}(\bar{y}),\quad \forall j\in J, \\& \lambda _{k}^{h}h_{k}(\bar{x})= \lambda _{k}^{h}h_{k}(\bar{y}),\quad \forall k\in I_{n}, \\& \lambda _{i}^{H}H_{i}(\bar{x})\leq \lambda _{i}^{H}H_{i}(\bar{y}),\quad \forall i\in I_{l}, \\& \lambda _{i}^{G}G_{i}(\bar{x})\leq \lambda _{i}^{G}G_{i}(\bar{y}),\quad \forall i\in I_{l} \end{aligned}$$

and by the definition of the index sets above, we get

$$ \textstyle\begin{cases} g_{j}(\bar{x})\leq g_{j}(\bar{y}), &\forall j\in I_{g}^{+}(\bar{x}), \\ h_{k}(\bar{x})=h_{k}(\bar{y}), &\forall k\in I_{h}^{+}(\bar{x}) \cup I_{h}^{-}(\bar{x}), \\ -H_{i}(\bar{x})\leq -H_{i}(\bar{y}), &\forall i\in I_{+}^{+}( \bar{x})\cup I_{00}^{+}(\bar{x})\cup I_{0-}^{+}(\bar{x})\cup I_{0+}^{+}( \bar{x}), \\ -H_{i}(\bar{x})\geq -H_{i}(\bar{y}), &\forall i\in I_{0+}^{-}( \bar{x}), \\ G_{i}(\bar{x})\leq G_{i}(\bar{y}),& \forall i\in \tilde{I}_{+}^{+}( \bar{x}). \end{cases} $$

(3.10)

By the $\partial _{c}$-quasiconvexity of the functions in assumption(ii) and (3.10), it follows that

$$\begin{aligned}& \bigl\langle \xi _{j}^{g},\bar{x}-\bar{y}\bigr\rangle \leq 0, \qquad \lambda _{j}^{g}>0,\quad \forall \xi _{j}^{g}\in \partial _{c}g_{j}( \bar{y}), j\in I_{g}^{+}( \bar{x}), \\& \bigl\langle \xi _{k}^{h},\bar{x}-\bar{y}\bigr\rangle \leq 0,\qquad \lambda _{k}^{h}>0,\quad \forall \xi _{k}^{h}\in \partial _{c}h_{k}( \bar{y}), k\in I_{h}^{+}( \bar{x}), \\& \bigl\langle \xi _{k}^{h},\bar{x}-\bar{y}\bigr\rangle \geq 0,\qquad \lambda _{k}^{h}< 0,\quad \forall \xi _{k}^{h}\in \partial _{c}h_{k}( \bar{y}), k\in I_{h}^{-}( \bar{x}), \\& \bigl\langle -\xi _{i}^{H},\bar{x}-\bar{y}\bigr\rangle \leq 0,\qquad \lambda _{i}^{H} \geq 0,\quad \forall \xi _{i}^{H}\in \partial _{c}H_{i}( \bar{y}), i \in I_{+}^{+}(\bar{x})\cup I_{00}^{+}(\bar{x})\cup I_{0-}^{+}( \bar{x}) \cup I_{0+}^{+}(\bar{x}), \\& \bigl\langle -\xi _{i}^{H},\bar{x}-\bar{y}\bigr\rangle \geq 0,\qquad \lambda _{i}^{H} \leq 0,\quad \forall \xi _{i}^{H}\in \partial _{c}H_{i}( \bar{y}), i \in I_{0+}^{-}(\bar{x}), \\& \bigl\langle \xi _{i}^{G},\bar{x}-\bar{y}\bigr\rangle \leq 0,\qquad \lambda _{i}^{G} \geq 0,\quad \forall \xi _{i}^{G}\in \partial _{c}G_{i}( \bar{y}), i \in \tilde{I}_{+}^{+}(\bar{x}) \end{aligned}$$

that is,

$$ \Biggl\langle \sum_{j\in J}\lambda _{j}^{g}\xi _{j}^{g}+ \sum _{k=1}^{n}\lambda _{k}^{h} \xi _{k}^{h} -\sum_{i=1}^{l} \lambda _{i}^{H}\xi _{i}^{H}+ \sum_{i=1}^{l}\lambda _{i}^{G} \xi _{i}^{G}, \bar{x}- \bar{y} \Biggr\rangle \leq 0. $$

By the above inequality and $0\in \Delta (\bar{y})$, there exist $\xi ^{L}\in \partial _{c}F^{L}(\bar{y})$ and $\xi ^{U}\in \partial _{c}F^{U}(\bar{y})$ such that

$$ \bigl\langle \alpha ^{L} \xi ^{L}+\alpha ^{U}\xi ^{U}, \bar{x}-\bar{y}\bigr\rangle \geq 0. $$

(3.11)

By (3.7) and the $\partial _{c}$-pseudoconvexity of $F^{L}(\cdot )$ and $F^{U}(\cdot )$, it follows that

$$\begin{aligned}& \bigl\langle \xi ^{L},\bar{x}-\bar{y}\bigr\rangle < 0, \quad \forall \xi ^{L}\in \partial _{c}F^{L}(\bar{y}), \\& \bigl\langle \xi ^{U},\bar{x}-\bar{y}\bigr\rangle < 0, \quad \forall \xi ^{U}\in \partial _{c}F^{U}(\bar{y}), \end{aligned}$$

then $\langle \alpha ^{L}\xi ^{L}+\alpha ^{U}\xi ^{U}, \bar{x}-\bar{y} \rangle <0$, $\alpha ^{L}, \alpha ^{U}\in R_{+}$, $\alpha ^{L}+\alpha ^{U}=1$, contradicting (3.11), so the result also holds. □

Theorem 3.5

(Restricted converse duality)

Let $\bar{x}\in E$ be a feasible point of (IOPVC), and let $(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{W}$ be a feasible point of $(D_{W})$ such that $F(\bar{x})=\Phi (y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$. If $\Phi ^{L}(\cdot , \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$, $\Phi ^{U}(\cdot ,\alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$ are $\partial _{c}$-pseudoconvex at $y\in E\cup \mathit{prE}_{W}$, then x̄ is the locally weakly LU optimal solution of (IOPVC).

Proof

Suppose that x̄ is not a locally weakly LU optimal solution of (IOPVC), then there exists $\tilde{x}\in E\cap B(\bar{x};\delta )$ such that $F(\tilde{x})<_{\mathit{LU}}^{s}F(\bar{x})$. By $F(\bar{x})=\Phi (y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$, we get $F(\tilde{x})<_{\mathit{LU}}^{s}\Phi (y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$, contradicting Theorem 3.1. So, x̄ 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:

$$\begin{aligned} \text{(IOPVC1)}\quad \min \quad& F(x)=\bigl[F^{L}(x), F^{U}(x)\bigr]=\bigl[x_{1}^{2}-x_{2}^{2}, x_{1}^{2}\bigr] \\ \text{s.t.}\quad &g_{1}(x)=-x_{1} \leq 0 , \\ &H_{1}(x)=x_{1}-x_{2}\geq0 , \\ &G_{1}(x)H_{1}(x)=x_{1}(x_{1}-x_{2})\leq 0. \end{aligned}$$

The feasible set of problem (IOPVC1) is given by

$$ E_{1}:=\{x\in R\mid x_{1}>0,x_{1}-x_{2}=0 \}\cup \{x\in R\mid x_{1}=0,x_{2} \leq 0\}. $$

For any $x\in E_{1}$, the Wolfe type dual model to (IOPVC1) is given by

$$\begin{aligned} \bigl(D_{W}(x)\bigr) \quad \max\quad &\Phi \bigl(y, \alpha ^{L}, \alpha ^{U}, \lambda _{1}^{g}, \lambda _{1}^{H}, \lambda _{1}^{G} \bigr) \\ \text{s.t.}\quad &\alpha ^{L}(2y_{1}, -2y_{2})+\alpha ^{U}(2y_{1}, 0)+\lambda _{1}^{g}(-1,0) -\lambda _{1}^{H}(1,-1)+ \lambda _{1}^{G}(1,0)=(0,0), \\ &\lambda _{1}^{H}\geq 0,\quad \text{if } 1\in I_{+}(x)\cup I_{0-}(x)\cup I_{00}(x); \\ &\lambda _{1}^{H}\in R,\quad \text{if } 1\in I_{0+}(x), \\ &\lambda _{1}^{G}\geq 0,\quad \text{if } 1\in I_{+}(x);\qquad \lambda _{1}^{G}=0 \quad \text{if } 1\in I_{0}(x), \end{aligned}$$

where $\Phi (y, \alpha ^{L}, \alpha ^{U}, \lambda _{1}^{g}, \lambda _{1}^{H}, \lambda _{1}^{G}) =[y_{1}^{2}-y_{2}^{2}-\lambda _{1}^{g}y_{1}- \lambda _{1}^{H}(y_{1}-y_{2}) +\lambda _{1}^{G}y_{1}, y_{1}^{2}- \lambda _{1}^{g}y_{1}-\lambda _{1}^{H}(y_{1}-y_{2}) +\lambda _{1}^{G}y_{1}]$. Therefore, we can get the feasible set of problem $(D_{W})$, which is not dependent on x,

$$\begin{aligned} E_{2}:={}&\bigl\{ \bigl(y_{1}, y_{2}, \alpha ^{L}, \alpha ^{U}, \lambda _{1}^{g}, \lambda _{1}^{H}, \lambda _{1}^{G} \bigr): 2y_{1}-\lambda _{1}^{g}-\lambda _{1}^{H}+ \lambda _{1}^{G}=0, -2\alpha ^{L}y_{2}+\lambda _{1}^{H}=0, \\ &{}y_{1}, y_{2}\in X, \alpha ^{L}, \alpha ^{U}\in R_{+}, \alpha ^{L}+ \alpha ^{U}=1, \lambda _{1}^{g}\geq 0, \lambda _{1}^{G}=0, \lambda _{1}^{H} \geq 0 \bigr\} . \end{aligned}$$

Let $\alpha ^{L}=\alpha ^{U}=\frac{1}{2}$, $\lambda _{1}^{g}=\lambda _{1}^{H}= \beta$ ($\beta \geq 0$), one has $y_{1}=\beta$, $y_{2}=\beta$.

And by $F(x)=\Phi (y, \alpha ^{L}, \alpha ^{U}, \lambda _{1}^{g}, \lambda _{1}^{H}, \lambda _{1}^{G})$, we get

$$\begin{aligned}& F^{L}(x)=\Phi ^{L}\bigl(y, \alpha ^{L}, \alpha ^{U}, \lambda _{1}^{g}, \lambda _{1}^{H}, \lambda _{1}^{G} \bigr)=-\beta ^{2}\leq 0\quad \Rightarrow\quad x_{1}^{2}-x_{2}^{2} \leq 0, \\& F^{U}(x)=\Phi ^{U}\bigl(y, \alpha ^{L}, \alpha ^{U}, \lambda _{1}^{g}, \lambda _{1}^{H}, \lambda _{1}^{G} \bigr)=0\quad \Rightarrow\quad x_{2}^{2}=0, \end{aligned}$$

that is, $x=(0,0)$. By $\lambda _{1}^{g}g_{1}(x)\leq 0$, $\lambda _{1}^{G}G_{1}(x)=0$, $- \lambda _{1}^{H}H_{1}(x)\leq 0$, and the $\partial _{c}$-pseudoconvexity of $\Phi ^{L}(\cdot , \alpha ^{L}, \alpha ^{U}, \lambda _{1}^{g}, \lambda _{1}^{H}, \lambda _{1}^{G})$ and $\Phi ^{U}(\cdot ,\alpha ^{L}, \alpha ^{U}, \lambda _{1}^{g}, \lambda _{1}^{H}, \lambda _{1}^{G})$ at $y\in E_{1}\cup prE_{2}$, we get $x=(0,0)$ is the locally weakly LU optimal solution of (IOPVC1).

Theorem 3.6

(Strict converse duality)

Let $\bar{x}\in E$ be the locally weakly LU optimal solution of (IOPVC) such that the (VC-ACQ) holds at x̄ and △ is closed. Assume that the conditions of Theorem 3.3hold and $(\bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{W}(\bar{x})$ is the locally weakly LU optimal solution of $(D_{W}(\bar{x}))$. If $\Phi ^{L}(\cdot , \alpha ^{L},\alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$, $\Phi ^{U}(\cdot , \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$ are strictly $\partial _{c}$-pseudoconvex at $y\in E\cup prE_{W}(\bar{x})$, then $\bar{x}=\bar{y}$.

Proof

Suppose that $\bar{x}\neq\bar{y}$. By Theorem 3.3, there exist Lagrange multipliers $\bar{\alpha }^{L}, \bar{\alpha }^{U}\in R_{+}$, $\bar{\lambda }^{g}\in \kappa (\bar{x})$, $\bar{\lambda }^{h}\in R^{n}$, $\bar{\lambda }^{H}, \bar{\lambda }^{G}, \bar{\varrho }, \bar{v}\in R^{l}$ such that $(\bar{x}, \bar{\alpha }^{L}, \bar{\alpha }^{U}, \bar{\lambda }^{g}, \bar{\lambda }^{h}, \bar{\lambda }^{H}, \bar{\lambda }^{G}, \bar{\varrho }, \bar{v})$ is the locally LU optimal solution of $(D_{W}(\bar{x}))$, that is,

$$ F(\bar{x})=\Phi \bigl(\bar{x}, \bar{\alpha }^{L}, \bar{\alpha }^{U}, \bar{\lambda }^{g}, \bar{\lambda }^{h}, \bar{\lambda }^{H}, \bar{\lambda }^{G} \bigr)=\Phi \bigl(\bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr). $$

Since $\bar{x}\in E$, $(\bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{W}( \bar{x})$, one has

$$\begin{aligned}& g_{j}(\bar{x})< 0,\qquad \lambda _{j}^{g} \geq 0,\quad j\notin I_{g}(\bar{x}), \\& g_{j}(\bar{x})=0,\qquad \lambda _{j}^{g} \geq 0,\quad j\in I_{g}(\bar{x}) , \\& h_{k}(\bar{x})=0,\qquad \lambda _{k}^{h} \in R,\quad k\in I_{n} , \\& -H_{i}(\bar{x})< 0,\qquad \lambda _{i}^{H} \geq 0,\quad i\in I_{+}(\bar{x}) , \\& -H_{i}(\bar{x})=0,\qquad \lambda _{i}^{H} \in R,\quad i\in I_{0}(\bar{x}) , \\& G_{i}(\bar{x})>0,\qquad \lambda _{i}^{G}=0, \quad i\in I_{0+}(\bar{x}), \\& G_{i}(\bar{x})=0,\qquad \lambda _{i}^{G} \geq 0,\quad i\in I_{00}(\bar{x}) \cup I_{+0}(\bar{x}), \\& G_{i}(\bar{x})< 0,\qquad \lambda _{i}^{G} \geq 0,\quad i\in I_{0-}(\bar{x})\cup I_{+-}(\bar{x}), \end{aligned}$$

and

$$ \sum_{j\in J}\lambda _{j}^{g}g_{j}( \bar{x})+\sum_{k=1}^{n} \lambda _{k}^{h}h_{k}(\bar{x})- \sum _{i=1}^{l}\lambda _{i}^{H}H_{i}( \bar{x})+\sum_{i=1}^{l}\lambda _{i}^{G}G_{i}(\bar{x})\leq 0. $$

And then we get

$$\begin{aligned}& \Phi ^{L}\bigl(\bar{x}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr)\leq \Phi ^{L}\bigl( \bar{y}, \alpha ^{L}, \alpha ^{U},\lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr), \\& \Phi ^{U}\bigl(\bar{x}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr)\leq \Phi ^{U}\bigl( \bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}\bigr). \end{aligned}$$

By the strict $\partial _{c}$-pseudoconvexity of $\Phi ^{L}(\cdot , \alpha ^{L}, \alpha ^{U},\lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$ and $\Phi ^{U}(\cdot , \alpha ^{L},\alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G})$ at $\bar{y}\in E\cup \mathit{prE}_{W}(\bar{x})$, we get (3.9), where $\xi ^{L}\in \partial _{c}F^{L}(\bar{y})$, $\xi ^{U}\in \partial _{c}F^{U}( \bar{y})$, $\xi _{j}^{g}\in \partial _{c}g_{j}(\bar{y})$, $j\in J$, $\xi _{k}^{h}\in \partial _{c}h_{k}(\bar{y})$, $k\in I_{n}$, $\xi _{i}^{H} \in \partial _{c}H_{i}(\bar{y})$, $i\in I_{l}$, $\xi _{i}^{G}\in \partial _{c}G_{i}(\bar{y})$, $i\in I_{l}$.

Combining (3.9) with $\alpha ^{L}$ and $\alpha ^{U}$, one has

$$ \langle \xi , \bar{x}-\bar{y}\rangle < 0,\quad \forall \xi \in \Delta (\bar{y}), $$

contradicting $0\in \Delta (\bar{y})$. □

4 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$(\mathit{VC}-\mathit{MWD}(x))$ in [6]: for $x\in E$,

$$\begin{aligned}& \bigl(D_{\mathit{MW}}(x)\bigr)\quad \mathit{LU}-\max \quad F(y) \\& \quad \text{s.t. } \textstyle\begin{cases} 0\in \Delta (y) \\ \alpha ^{L}, \alpha ^{U}\in R_{+},\quad \alpha ^{L}+\alpha ^{U}=1, \\ \lambda _{j}^{g}\geq 0, \qquad \lambda _{j}^{g}g_{j}(y)\geq 0, \quad \forall j \in J, \\ \lambda _{k}^{h}\in R, \qquad \lambda _{k}^{h}h_{k}(y)=0, \quad \forall k\in I_{n}, \\ \lambda _{i}^{G}G_{i}(y)\geq 0,\qquad \lambda _{i}^{G}=v_{i}H_{i}(x),\quad v_{i} \geq 0, \forall i\in I_{l}, \\ -\lambda _{i}^{H}H_{i}(y)\geq 0,\qquad \lambda _{i}^{H}=\varrho _{i}-v_{i}G_{i}(x), \quad \varrho _{i}\geq 0, \forall i\in I_{l}. \end{cases}\displaystyle \end{aligned}$$

(4.1)

Let $E_{\mathit{MW}}(x)$ denote the feasible set of $(D_{\mathit{MW}}(x))$, $\mathit{prE}_{\mathit{MW}}(x):=\{y\in X:(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{\mathit{MW}}(x)\}$ represents the projection of the set $E_{\mathit{MW}}(x)$ on X.

In order to be independent of (IOPVC), we give the another Mond–Weir type dual model:

$$\begin{aligned}& (D_{\mathit{MW}}) \quad \mathit{LU}\text{-}\max F(y) \\& \quad \text{s.t.} \bigl(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v\bigr)\in E_{\mathit{MW}}:=\bigcap_{x\in E}E_{\mathit{MW}}(x), \end{aligned}$$

where $E_{\mathit{MW}}$ denotes the set of all feasible points of $(D_{\mathit{MW}})$ and $\mathit{prE}_{\mathit{MW}}$ denotes the projection of the set $E_{\mathit{MW}}$ on X.

Definition 4.1

([4])

Let $x\in E$,

• (i) $(\bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{\mathit{MW}}(x)$ is said to be a locally LU optimal solution of $(D_{\mathit{MW}}(x))$ if there exists $B(\bar{y};\delta )$ such that there is no $y\in E_{\mathit{MW}}(x)\cap B(\bar{y};\delta )$ satisfying

  $$ F(\bar{y})< _{\mathit{LU}}F(y). $$

• (ii) $(\bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{\mathit{MW}}(x)$is said to be a locally weakly LU optimal solution of $(D_{\mathit{MW}}(x))$ if there exists $B(\bar{y};\delta )$ such that there is no $y\in E_{\mathit{MW}}(x)\cap B(\bar{y};\delta )$ satisfying

  $$ F(\bar{y})< _{\mathit{LU}}^{s}F(y). $$

Theorem 4.1

(Weak duality)

Let $x\in E$, $(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{\mathit{MW}}$ be feasible points for the (IOPVC) and the $(D_{\mathit{MW}})$, respectively. If one of the following conditions holds:

• (i) $F^{L}(\cdot )$, $F^{U}(\cdot )$ are $\partial _{c}$-pseudoconvex at $y\in E\cup \mathit{prE}_{\mathit{MW}}$, $\sum_{j\in J}\lambda _{j}^{g}g_{j}(\cdot )+\sum_{k=1}^{n} \lambda _{k}^{h}h_{k}(\cdot )- \sum_{i=1}^{l}\lambda _{i}^{H}H_{i}( \cdot )+\sum_{i=1}^{l}\lambda _{i}^{G}G_{i}(\cdot )$ is $\partial _{c}$-quasiconvex at $y\in E\cup \mathit{prE}_{\mathit{MW}}$;

• (ii) $F^{L}(\cdot )$, $F^{U}(\cdot )$ are $\partial _{c}$-pseudoconvex at $y\in E\cup \mathit{prE}_{\mathit{MW}}$, $g_{j}$ ($j\in I_{g}^{+}(x)$), $h_{k}$ ($k\in I_{h}^{+}(x)$), $-h_{k}$ ($k \in I_{h}^{-}(x)$), $-H_{i}$ ($i\in I_{+}^{+}(x)\cup I_{00}^{+}(x)\cup I_{0-}^{+}(x) \cup I_{0+}^{+}(x)$), $H_{i}$ ($i\in I_{0+}^{-}(x)$), $G_{i}$ ($i\in \tilde{I}_{+}^{+}(x)$) are $\partial _{c}$-quasiconvex at $y\in E\cup \mathit{prE}_{\mathit{MW}}$;

Then

$$ F(x)\nless_{\mathit{LU}}^{s}F(y). $$

Proof

Suppose that $F(x)<_{\mathit{LU}}^{s}F(y)$, there exists

$$ \bigl[F^{L}(x), F^{U}(x)\bigr]< _{\mathit{LU}}^{s} \bigl[F^{L}(y), F^{U}(y)\bigr]. $$

(4.2)

(i) Since $x\in E$, $(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{\mathit{MW}}$, one has (3.3). By (4.1) and (3.3), we get

$$\begin{aligned}& \sum_{j\in J}\lambda _{j}^{g}g_{j}(x)+ \sum_{k=1}^{n} \lambda _{k}^{h}h_{k}(x)- \sum _{i=1}^{l}\lambda _{i}^{H}H_{i}(x)+ \sum_{i=1}^{l}\lambda _{i}^{G}G_{i}(x) \\& \quad \leq \sum_{j\in J}\lambda _{i}^{g}g_{i}(y)+ \sum_{k=1}^{n} \lambda _{k}^{h}h_{k}(y)- \sum _{i=1}^{l}\lambda _{i}^{H}H_{i}(y)+ \sum_{i=1}^{l}\lambda _{i}^{G}G_{i}(y), \end{aligned}$$

and by the $\partial _{c}$-quasiconvexity of the above functions, one has

$$ \Biggl\langle \sum_{j\in J}\lambda _{j}^{g}\xi _{j}^{g}+ \sum _{k=1}^{n}\lambda _{k}^{h} \xi _{k}^{h} -\sum_{i=1}^{l} \lambda _{i}^{H}\xi _{i}^{H}+ \sum_{i=1}^{l}\lambda _{i}^{G} \xi _{i}^{G}, x-y \Biggr\rangle \leq 0, $$

where $\xi _{j}^{g}\in \partial _{c}g_{j}(y)$, $\xi _{k}^{h}\in \partial _{c}h_{k}(y)$, $\xi _{i}^{H}\in \partial _{c}H_{i}(y)$, $\xi _{i}^{G}\in \partial _{c}G_{i}(y)$. Using the above inequality and $0\in \Delta (y)$, there exist $\xi ^{L}\in \partial _{c}F^{L}(y)$ and $\xi ^{U}\in \partial _{c}F^{U}(y)$ such that

$$ \bigl\langle \alpha ^{L} \xi ^{L}+\alpha ^{U}\xi ^{U}, x-y\bigr\rangle \geq 0. $$

(4.3)

By (4.2) and the $\partial _{c}$-pseudoconvexity of $F^{L}(\cdot )$ and $F^{U}(\cdot )$, it follows that

$$\begin{aligned}& \bigl\langle \xi ^{L},x-y\bigr\rangle < 0,\quad \forall \xi ^{L}\in \partial _{c}F^{L}(y), \\& \bigl\langle \xi ^{U},x-y\bigr\rangle < 0,\quad \forall \xi ^{U}\in \partial _{c}F^{U}(y). \end{aligned}$$

Then $\langle \alpha ^{L}\xi ^{L}+\alpha ^{U}\xi ^{U}, x-y\rangle <0$, $\alpha ^{L}, \alpha ^{U}\in R_{+}$, $\alpha ^{L}+\alpha ^{U}=1$, which contradicts (4.3).

(ii) By (4.1) and (3.3), one has

$$ \textstyle\begin{cases} g_{j}(x)\leq g_{j}(y), &\forall j\in I_{g}^{+}(x), \\ h_{k}(x)=h_{k}(y), &\forall k\in I_{h}^{+}(x)\cup I_{h}^{-}(x), \\ -H_{i}(x)\leq -H_{i}(y), &\forall i\in I_{+}^{+}(x)\cup I_{00}^{+}(x) \cup I_{0-}^{+}(x)\cup I_{0+}^{+}(x), \\ -H_{i}(x)\geq -H_{i}(y), &\forall i\in I_{0+}^{-}(x), \\ G_{i}(x)\leq G_{i}(y),& \forall i\in \tilde{I}_{+}^{+}(x). \end{cases} $$

(4.4)

Combining (4.4) with the $\partial _{c}$-quasiconvexity of the above functions, we get

$$\begin{aligned}& \bigl\langle \xi _{j}^{g},x-y\bigr\rangle \leq 0,\qquad \lambda _{j}^{g}>0,\quad \forall \xi _{j}^{g}\in \partial _{c}g_{j}(y), j\in I_{g}^{+}(x), \\& \bigl\langle \xi _{k}^{h},x-y\bigr\rangle \leq 0,\qquad \lambda _{k}^{h}>0,\quad \forall \xi _{k}^{h}\in \partial _{c}h_{k}(y), k\in I_{h}^{+}(x), \\& \bigl\langle \xi _{k}^{h},x-y\bigr\rangle \geq 0,\qquad \lambda _{k}^{h}< 0, \quad \forall \xi _{k}^{h}\in \partial _{c}h_{k}(y), k\in I_{h}^{-}(x), \\& \bigl\langle -\xi _{i}^{H},x-y\bigr\rangle \leq 0, \qquad \lambda _{i}^{H}\geq 0,\quad \forall \xi _{i}^{H}\in \partial _{c}H_{i}(y), i\in I_{+}^{+}(x)\cup I_{00}^{+}(x) \cup I_{0-}^{+}(x)\cup I_{0+}^{+}(x), \\& \bigl\langle -\xi _{i}^{H},x-y\bigr\rangle \geq 0, \qquad \lambda _{i}^{H}\leq 0,\quad \forall \xi _{i}^{H}\in \partial _{c}H_{i}(y), i\in I_{0+}^{-}(x), \\& \bigl\langle \xi _{i}^{G},x-y\bigr\rangle \leq 0,\qquad \lambda _{i}^{G}\geq 0,\quad \forall \xi _{i}^{G}\in \partial _{c}G_{i}(y), i\in \tilde{I}_{+}^{+}(x) \end{aligned}$$

that is,

$$ \Biggl\langle \sum_{j\in J}\lambda _{j}^{g}\xi _{j}^{g}+ \sum _{k=1}^{n}\lambda _{k}^{h} \xi _{k}^{h} -\sum_{i=1}^{l} \lambda _{i}^{H}\xi _{i}^{H}+ \sum_{i=1}^{l}\lambda _{i}^{G} \xi _{i}^{G}, x-y \Biggr\rangle \leq 0. $$

The rest is proved to be the same as the latter part of (i). □

Theorem 4.2

(Weak duality)

Let $x\in E$, $(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{\mathit{MW}}$ be feasible points for the (IOPVC) and the $(D_{\mathit{MW}})$, respectively. If one of the following conditions holds:

• (i) $F^{L}(\cdot )$, $F^{U}(\cdot )$ are strictly $\partial _{c}$-pseudoconvex at $y\in E\cup \mathit{prE}_{\mathit{MW}}$, $\sum_{j\in J}\lambda _{j}^{g}g_{j}(\cdot )+\sum_{k=1}^{n} \lambda _{k}^{h}h_{k}(\cdot )- \sum_{i=1}^{l}\lambda _{i}^{H}H_{i}( \cdot )+\sum_{i=1}^{l}\lambda _{i}^{G}G_{i}(\cdot )$ is $\partial _{c}$-quasiconvex at $y\in E\cup \mathit{prE}_{\mathit{MW}}$;

• (ii) $F^{L}(\cdot )$, $F^{U}(\cdot )$ are strictly $\partial _{c}$-pseudoconvex at $y\in E\cup \mathit{prE}_{\mathit{MW}}$, $g_{j}$ ($j\in I_{g}^{+}(x)$), $h_{k}$ ($k\in I_{h}^{+}(x)$), $-h_{k}$ ($k \in I_{h}^{-}(x)$), $-H_{i}$ ($i\in I_{+}^{+}(x)\cup I_{00}^{+}(x)\cup I_{0-}^{+}(x) \cup I_{0+}^{+}(x)$), $H_{i}$ ($i\in I_{0+}^{-}(x)$), $G_{i}$ ($i\in \tilde{I}_{+}^{+}(x)$) are $\partial _{c}$-quasiconvex at $y\in E\cup \mathit{prE}_{\mathit{MW}}$;

Then

$$ F(x)\nless_{\mathit{LU}}F(y). $$

Proof

The proof is similar to Theorem 4.1. □

Theorem 4.3

(Strong duality)

Let $\bar{x}\in E$ be the locally weakly LU optimal solution of (IOPVC), and the condition (VC-ACQ) holds at x̄ and △ is closed. Then there exist $\bar{\alpha }^{L}, \bar{\alpha }^{U}\in R_{+}$, $\bar{\lambda }^{g}\in R_{+}^{|J|}$, $\bar{\lambda }^{h}\in R^{n}$, $\bar{\lambda }^{H}, \bar{\lambda }^{G}, \bar{\varrho }, \bar{v}\in R^{l}$ such that $(\bar{x}, \bar{\alpha }^{L}, \bar{\alpha }^{U}, \bar{\lambda }^{g}, \bar{\lambda }^{h}, \bar{\lambda }^{H}, \bar{\lambda }^{G}, \bar{\varrho }, \bar{v})$ is a feasible point of $(D_{\mathit{MW}}(\bar{x}))$. Moreover, if the hypotheses of Theorem 4.1hold, then $(\bar{x}, \bar{\alpha }^{L}, \bar{\alpha }^{U}, \bar{\lambda }^{g}, \bar{\lambda }^{h}, \bar{\lambda }^{H}, \bar{\lambda }^{G}, \bar{\varrho }, \bar{v})$ is the locally weakly LU optimal solution of $(D_{\mathit{MW}}(\bar{x}))$.

Proof

Since x̄ is the locally weakly LU optimal solution of (IOPVC) and (VC-ACQ) holds at x̄, so by Theorem 2.1, there exist $\bar{\alpha }^{L}, \bar{\alpha }^{U}\in R_{+}$, $\bar{\lambda }^{g}\in \kappa (\bar{x})$, $\bar{\lambda }^{h}\in R^{n}$, $\bar{\lambda }^{H}, \bar{\lambda }^{G}, \bar{\varrho }, \bar{v}\in R^{l}$ such that (2.3) and (2.4) hold. From this together with the definition of $(D_{\mathit{MW}}(\bar{x}))$, one has $(\bar{x}, \bar{\alpha }^{L}, \bar{\alpha }^{U}, \bar{\lambda }^{g}, \bar{\lambda }^{h}, \bar{\lambda }^{H}, \bar{\lambda }^{G}, \bar{\varrho }, \bar{v})$ is a feasible point of $(D_{\mathit{MW}}(\bar{x}))$. By Theorem 4.1, we know

$$ F(\bar{x})\nless_{\mathit{LU}}^{s}F(y), \forall \bigl(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v \bigr) \in E_{\mathit{MW}}(\bar{x}), $$

so $(\bar{x}, \bar{\alpha }^{L}, \bar{\alpha }^{U}, \bar{\lambda }^{g}, \bar{\lambda }^{h}, \bar{\lambda }^{H}, \bar{\lambda }^{G}, \bar{\varrho }, \bar{v})$ is the locally weakly LU optimal solution of $(D_{\mathit{MW}}(\bar{x}))$. □

Theorem 4.4

(Converse duality)

Let $x\in E$ be any feasible point of (IOPVC) and $(\bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{\mathit{MW}}$ be a feasible point of $(D_{\mathit{MW}})$. If one of the following conditions holds:

• (i) $F^{L}(\cdot )$, $F^{U}(\cdot )$ are $\partial _{c}$-pseudoconvex at $\bar{y}\in E\cup \mathit{prE}_{\mathit{MW}}$, $\sum_{j\in J}\lambda _{j}^{g}g_{j}(\cdot )+\sum_{k=1}^{n} \lambda _{k}^{h}h_{k}(\cdot )- \sum_{i=1}^{l}\lambda _{i}^{H}H_{i}( \cdot )+\sum_{i=1}^{l}\lambda _{i}^{G}G_{i}(\cdot )$ is $\partial _{c}$-quasiconvex at $\bar{y}\in E\cup \mathit{prE}_{\mathit{MW}}$;

• (ii) $F^{L}(\cdot )$, $F^{U}(\cdot )$ are $\partial _{c}$-pseudoconvex at $\bar{y}\in E\cup \mathit{prE}_{\mathit{MW}}$, $g_{j}$ ($j\in I_{g}^{+}(x)$), $h_{k}$ ($k\in I_{h}^{+}(x)$), $-h_{k}$ ($k \in I_{h}^{-}(x)$), $-H_{i}$ ($i\in I_{+}^{+}(x)\cup I_{00}^{+}(x)\cup I_{0-}^{+}(x) \cup I_{0+}^{+}(x)$), $H_{i}$ ($i\in I_{0+}^{-}(x)$), $G_{i}$ ($\tilde{I}_{+}^{+}(x)$) are $\partial _{c}$-quasiconvex at $\bar{y}\in E\cup \mathit{prE}_{\mathit{MW}}$;

Then ȳ is the locally weakly LU optimal solution of (IOPVC).

Proof

Suppose to the contrary that ȳ is not the locally weakly LU optimal solution of (IOPVC), then one has (3.7).

(i) Since $\bar{x}\in E$ and $(\bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)$ are feasible points for the (IOPVC) and the $(D_{\mathit{MW}})$, by (4.1) and (3.3), we know that (3.8) holds. By the $\partial _{c}$-quasiconvexity of $\sum_{j\in J}\lambda _{j}^{g}g_{j}(\cdot )+\sum_{k=1}^{n} \lambda _{k}^{h}h_{k}(\cdot )- \sum_{i=1}^{l}\lambda _{i}^{H}H_{i}( \cdot )+\sum_{i=1}^{l}\lambda _{i}^{G}G_{i}(\cdot )$ at $\bar{y}\in E\cup \mathit{prE}_{\mathit{MW}}$, we get

$$ \Biggl\langle \sum_{j\in J}\lambda _{j}^{g}\xi _{j}^{g}+ \sum _{k=1}^{n}\lambda _{k}^{h} \xi _{k}^{h} -\sum_{i=1}^{l} \lambda _{i}^{H}\xi _{i}^{H}+ \sum_{i=1}^{l}\lambda _{i}^{G} \xi _{i}^{G}, \bar{x}- \bar{y} \Biggr\rangle \leq 0, $$

where $\xi _{j}^{g}\in \partial _{c}g_{j}(\bar{y})$, $\xi _{k}^{h}\in \partial _{c}h_{k}(\bar{y})$, $\xi _{i}^{H}\in \partial _{c}H_{i}( \bar{y})$, $\xi _{i}^{G}\in \partial _{c}G_{i}(\bar{y})$.

Combining this with $0\in \Delta (\bar{y})$, one has (3.11). And by (3.7) and the $\partial _{c}$-pseudoconvexity of $F^{L}(\cdot )$ and $F^{U}(\cdot )$, one has

$$\begin{aligned}& \bigl\langle \xi ^{L},\bar{x}-\bar{y}\bigr\rangle < 0,\quad \forall \xi ^{L}\in \partial _{c}F^{L}(\bar{y}), \\& \bigl\langle \xi ^{U},\bar{x}-\bar{y}\bigr\rangle < 0, \quad \forall \xi ^{U}\in \partial _{c}F^{U}(\bar{y}) \end{aligned}$$

and $\langle \alpha ^{L}\xi ^{L}+\alpha ^{U}\xi ^{U}, \bar{x}-\bar{y} \rangle <0$, $\alpha ^{L}, \alpha ^{U}\in R_{+}$, $\alpha ^{L}+\alpha ^{U}=1$. This is a contradiction to (3.11), and hence the result is proved.

(ii) The proof of (ii) is similar to the proof of Theorem 3.4(ii), so it is omitted. □

Theorem 4.5

(Restricted converse duality)

Let $\bar{x}\in E$ be a feasible point of (IOPVC), and let $(y, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{\mathit{MW}}$ be a feasible point of $(D_{\mathit{MW}})$ such that $F(\bar{x})=F(y)$. If the hypotheses of Theorem 4.1hold, then x̄ is the locally weakly LU optimal solution of (IOPVC).

Proof

Suppose that x̄ is not a locally weakly LU optimal solution of (IOPVC). Then there exist $\tilde{x}\in E\cap B(\bar{x};\delta )$ such that $F(\tilde{x})<_{\mathit{LU}}^{s}F(\bar{x})$. By $F(\bar{x})=F(y)$, one has $F(\tilde{x})<_{\mathit{LU}}^{s}F(y)$, contradicting Theorem 4.1. □

Let us use the following example to show that the conclusion of Theorem 4.5 holds.

Example 4.1

Let $X=R^{2}$, $n=0$, $J=l=1$, we give the following problem:

$$\begin{aligned} \text{(IOPVC2)}\quad \min\quad & F(x)=\bigl[F^{L}(x), F^{U}(x)\bigr]=\bigl[x_{1}^{2}-x_{2}^{2}, x_{1}^{2}\bigr] \\ \text{s.t.}\quad &g_{1}(x)=-x_{1}\leq 0, \\ & H_{1}(x)=x_{1}-x_{2}\geq 0, \\ &G_{1}(x)H_{1}(x)=x_{1}(x_{1}-x_{2}) \leq 0. \end{aligned}$$

It is easy to know that

$$ E_{3}=\{x\in R\mid x_{1}>0,x_{1}-x_{2}=0 \}\cup \{x\in R\mid x_{1}=0,x_{2} \leq 0\} $$

is the feasible set of (IOPVC2). For any $x\in E_{3}$, the Mond–Weir type dual model to (IOPVC2) is given by

$$\begin{aligned} \bigl(D_{\mathit{MW}}(x)\bigr)\quad \max \quad & F(y)=\bigl[F^{L}(y), F^{U}(y)\bigr]=\bigl[y_{1}^{2}-y_{2}^{2}, y_{1}^{2}\bigr] \\ \text{s.t.}\quad &\alpha ^{L}(2y_{1}, -2y_{2})+\alpha ^{U}(2y_{1}, 0)+ \lambda _{1}^{g}(-1,0) \\ &-\lambda _{1}^{H}(1,-1)+\lambda _{1}^{G}(1,0)=(0,0) , \\ &\lambda _{1}^{g}\geq 0,\qquad \lambda _{1}^{g}g_{1}(y)\geq 0, \\ &\lambda _{1}^{H}\geq 0,\quad \text{if } 1\in I_{+}(x)\cup I_{0-}(x)\cup I_{00}(x), \\ &\lambda _{1}^{H}\in R,\quad \text{if } 1\in I_{0+}(x),\qquad -\lambda _{1}^{H}H_{1}(y) \geq 0, \\ &\lambda _{1}^{G}\geq 0,\quad \text{if } 1\in I_{+}(x),\qquad \lambda _{1}^{G}=0,\quad \text{if } 1\in I_{0}(x),\qquad \lambda _{1}^{G}G_{1}(y) \geq 0. \end{aligned}$$

Therefore, we can get the feasible set of problem $(D_{\mathit{MW}})$, which is not dependent on x,

$$\begin{aligned} E_{4}:={}&\bigl\{ \bigl(y_{1}, y_{2}, \alpha ^{L}, \alpha ^{U}, \lambda _{1}^{g}, \lambda _{1}^{H}, \lambda _{1}^{G} \bigr): \\ &2y_{1}-\lambda _{1}^{g}-\lambda _{1}^{H}+\lambda _{1}^{G}=0, -2 \alpha ^{L}y_{2}+\lambda _{1}^{H}=0, \\ &y_{1}, y_{2}\in X, \alpha ^{L}, \alpha ^{U}\in R_{+}, \alpha ^{L}+ \alpha ^{U}=1, -\lambda _{1}^{g}y_{1} \geq 0, \\ &\lambda _{1}^{G}y_{1}\geq 0, -\lambda _{1}^{H}(y_{1}-y_{2})\geq 0, \lambda _{1}^{H}\geq 0, \lambda _{1}^{G}=0 \bigr\} . \end{aligned}$$

Let $\alpha ^{L}=\alpha ^{U}=\frac{1}{2}$, $\lambda _{1}^{g}=\lambda _{1}^{H}= \beta$ ($\beta \geq 0$), one has $y_{1}=\beta$, $y_{2}=\beta$.

By $F(x)=F(y)$, one has

$$\begin{aligned}& F^{L}(x)=F^{L}(y)=0\quad \Rightarrow\quad x_{1}^{2}-x_{2}^{2}=0, \\& F^{U}(x)=F^{U}(y)=\beta ^{2}\geq 0\quad \Rightarrow\quad x_{1}^{2}\geq 0. \end{aligned}$$

Then we get $\lambda _{1}^{g}g_{1}(x)\leq 0$, $\lambda _{1}^{G}G_{1}(x)=0$, $- \lambda _{1}^{H}H_{1}(x)\leq 0$. Finally, by the $\partial _{c}$-pseudoconvexity of $F^{L}(\cdot )$ and $F^{U}(\cdot )$ at $y\in E_{3}\cup \mathit{prE}_{4}$ and the $\partial _{c}$-quasiconvexity of $\lambda _{1}^{g}g_{1}(\cdot )-\lambda _{1}^{H}H_{1}(\cdot )+\lambda _{1}^{G}G_{1}( \cdot )$, we get $x=(0,0)$ is the locally weakly LU optimal solution of (IOPVC2).

Theorem 4.6

(Strict converse duality)

Let $\bar{x}\in E$ be a locally weakly LU optimal solution of (IOPVC) such that the (VC-ACQ) holds at x̄ and △ is closed. Assume that the conditions of Theorem 4.3hold and $(\bar{y}, \alpha ^{L}, \alpha ^{U}, \lambda ^{g}, \lambda ^{h}, \lambda ^{H}, \lambda ^{G}, \varrho , v)\in E_{\mathit{MW}}(\bar{x})$ is the locally weakly LU optimal solution of $(D_{\mathit{MW}}(\bar{x}))$. If one of the following conditions holds:

• (i) $F^{L}(\cdot )$, $F^{U}(\cdot )$ are strictly $\partial _{c}$-pseudoconvex at $\bar{y}\in E\cup \mathit{prE}_{\mathit{MW}}(\bar{x})$, $\sum_{j\in J}\lambda _{j}^{g}g_{j}(\cdot )+\sum_{k=1}^{n} \lambda _{k}^{h}h_{k}(\cdot )- \sum_{i=1}^{l}\lambda _{i}^{H}H_{i}( \cdot )+\sum_{i=1}^{l}\lambda _{i}^{G}G_{i}(\cdot )$ is $\partial _{c}$-quasiconvex at $\bar{y}\in E\cup \mathit{prE}_{\mathit{MW}}(\bar{x})$;

• (ii) $F^{L}(\cdot )$, $F^{U}(\cdot )$ are strictly $\partial _{c}$-pseudoconvex at $\bar{y}\in E\cup \mathit{prE}_{\mathit{MW}}(\bar{x})$, $g_{j}$ ($j\in I_{g}^{+}(\bar{x})$), $h_{k}$ ($k\in I_{h}^{+}(\bar{x})$), $-h_{k}$ ($k\in I_{h}^{-}(\bar{x})$), $-H_{i}$ ($i\in I_{+}^{+}(x)\cup I_{00}^{+}( \bar{x})\cup I_{0-}^{+}(\bar{x})\cup I_{0+}^{+}(\bar{x})$), $H_{i}$ ($i \in I_{0+}^{-}(\bar{x})$), $G_{i}$ ($\tilde{I}_{+}^{+}(\bar{x})$) are $\partial _{c}$-quasiconvex at $\bar{y}\in E\cup \mathit{prE}_{\mathit{MW}}(\bar{x})$; then $\bar{x}=\bar{y}$.

Proof

Suppose that $\bar{x}\neq\bar{y}$. By Theorem 4.3, there exist Lagrange multipliers $\bar{\alpha }^{L}, \bar{\alpha }^{U}\in R_{+}$, $\bar{\lambda }^{g}\in \kappa (\bar{x})$, $\bar{\lambda }^{h}\in R^{n}$, $\bar{\lambda }^{H}, \bar{\lambda }^{G}, \bar{\varrho }, \bar{v}\in R^{l}$ such that $(\bar{x}, \bar{\alpha }^{L}, \bar{\alpha }^{U}, \bar{\lambda }^{g}, \bar{\lambda }^{h}, \bar{\lambda }^{H}, \bar{\lambda }^{G}, \bar{\varrho }, \bar{v})$ is the locally weakly LU optimal solution of $(D_{\mathit{MW}}(\bar{x}))$, it follows that

$$ F(\bar{x})=F(\bar{y}). $$

(4.5)

The remaining parts are similar to (i) and (ii) of Theorem 4.4, so they are omitted. □

5 Concluding remarks

In this paper, we study the duality theorems of nondifferentiable semi-infinite interval-valued 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 $\partial _{c}$-pseudoconvex, strictly $\partial _{c}$-pseudoconvex, and $\partial _{c}$-quasiconvex.