Mathematical programs with vanishing constraints involving strongly invex functions

Abstract

In this paper, we study Wolfe and Mond–Weir type dual models for a special class of optimization problems known as the mathematical programs with vanishing constraints (MPVC). We establish the weak, strong, converse, restricted converse and strict converse duality results under the assumptions of generalized higher order invexity and strict invexity between the primal MPVC and the corresponding Wolfe type dual. We also derive the weak, strong, converse, restricted converse and strict converse duality results between the primal MPVC and the corresponding Mond–Weir type dual under the assumptions of generalized higher order pseudoinvex, strict pseudoinvex and quasiinvex functions.

1 Introduction

MPVC problem originates from the optimization topology design problems in mechanical structures [1]. Usually vanishing constraints violet standard constraint qualifications, like the Mangasarian–Fromovitz constraint qualification and the linear independence constraint qualification, whereas the Abadie constraint qualification is a too strong assumption for the MPVC. There are some situations in which the Guignard constraint qualification may hold, but neither it is easy to see whether a given MPVC satisfies the Guignard constraint qualification, nor it is enough to prove convergence results for suitable algorithms. At present, the corresponding research shows that the robot motion planning problem [14] can be transformed into the MPVC problem. In addition, it is also widely used in the nonlinear integer optimal control [19] and the economic dispatch problem [10]. Several theoretical properties and different numerical approaches for MPVC can be found in [1, 6, 8, 9, 14, 15, 23].

Usually, generalized convex functions [12, 13] have been introduced in order to weaken the convexity requirements as much as possible to obtain results related to optimization theory. One of the significant generalization of convex function is invex function [4, 5]. The class of invex functions preserves many properties of the class of convex functions and has shown to be very useful in a variety of applications [22]. It is well known that optimality and duality theory provides the foundation of algorithms for a solution of an optimization problem and hence constitutes an important portion in the study of mathematical programming. Duality is very important in optimization as the weak duality provides a lower bound to the objective function value of the primal problem. The classical Wolfe duality was introduced by Wolfe [24], while for differentiable scalar functions the Mond–Weir duality was introduced in [20]. Later, these duality models were extended to nondifferentiable functions by utilizing different generalizations of the notion of convexity for both scalar and vector cases (See [2, 3, 16, 17, 21]). Recently, Mishra et al. [21] and Hu et al. [7] formulate and study Wolfe and Mond–Weir type dual models for the mathematical programs with vanishing constraints. They establish the weak, strong, converse, restricted converse and strict converse duality results between the primal mathematical programs with vanishing constraints and the corresponding dual models under some assumptions.

Motivated by the work of Mishra et al. [21] and Hu et al. [7], we establish some duality results for generalized invex functions for a differentiable framework. The outline of this paper is as follows: in Section 2, we give some preliminaries about the MPVC. In Section 3, we give the new Wolfe and Mond–Weir type dual models for MPVC and some duality results. We close with final remarks and some future work in Section 4.

2 Preliminaries

The general form of mathematical program with vanishing constraints (MPVC) is as follows:

$$ \begin{array}{@{}rcl@{}} \text{min} \ \ \ \ &&f (x) \\ \text{subject to} \ &&g_{i}(x) \leqslant 0, i \in M, \ M := \left\lbrace 1, 2, . . . , m \right\rbrace,\\ &&h_{i}(x) = 0, i \in P, \ P := \left\lbrace 1, 2, . . . , p \right\rbrace,\\ &&H_{i}(x) \geqslant 0, i \in L, \ L := \left\lbrace 1, 2, . . . , l \right\rbrace,\\ &&G_{i}(x)H_{i}(x) \leqslant 0, i \in L, \ L := \left\lbrace 1, 2, . . . , l \right\rbrace, \end{array} $$

where $f : \mathbb {R}^{n} \to \mathbb {R}$ is Lipschitz continuous, $g=(g_{1},g_{2},...,g_{m}) : \mathbb {R}^{n} \to \mathbb {R}^{m},$ $h=(h_{1},h_{2},...,h_{p}) : \mathbb {R}^{n} \to \mathbb {R}^{p}, G=(G_{1},G_{2},...,G_{l}) : \mathbb {R}^{n} \to \mathbb {R}^{l}$, and $H=(H_{1},H_{2},...,H_{l}) : \mathbb {R}^{n} \to \mathbb {R}^{l}$ are all continuously differentiable functions. Throughout this paper, X denotes the feasible region of MPVC and defined as

$$ \begin{array}{@{}rcl@{}} X:= \{x \in \mathbb{R}^{n}: &&g_{i}(x) \leqslant 0, \ i \in M,\\ &&h_{i}(x) = 0, \ i \in P,\\ &&H_{i}(x) \geqslant 0, \ i \in L,\\ &&G_{i}(x)H_{i}(x) \leqslant 0, \ i \in L\}. \end{array} $$

Definition 2.1

An n-dimensional open ball of radius r is the collection of points of distance less than r from a fixed point in Euclidean n-space. Explicitly, the open ball with center x and radius r is defined by

$$\mathbb{B}_{r}(x)=\{y: \vert y-x \vert < r\}.$$

The open ball for n = 1 is called an open interval, and the term open disk is sometimes used for n = 2 and sometimes as a synonym for open ball.

Definition 2.2

A point x^∗∈ X is said to be a local minimum of the MPVC, if and only if there exists an open ball B(x^∗,r) around x^∗ with radius r > 0 such that

$$f(x^{*}) \leq f(x), \forall x \in X \cap B(x^{*}, r).$$

A point x^∗∈ X is said to be a global minimum of the MPVC, if and only if

$$f(x^{*}) \leq f(x), \forall x \in X.$$

Let x^∗∈ X be any feasible point of the MPVC. The following index sets will be used in the sequel.

$$ \begin{array}{@{}rcl@{}} &{}l_{g}(x^{*})=\{i \vert g_{i}(x^{*})=0\}, &\\ &{\kern-4.5pc} l_{h}(x^{*})=\{1,2, \dots,p\},&\\ &{\kern-3.8pc}l_{+}(x^{*})=\{i \vert H_{i}(x^{*})>0\},&\\ &{\kern-3.8pc}l_{0}(x^{*})=\{i \vert H_{i}(x^{*})=0\},&\\ &l_{+0}(x^{*})=\{i \vert H_{i}(x^{*})>0,G_{i}(x^{*})=0\},&\\ &l_{+-}(x^{*})=\{i \vert H_{i}(x^{*})>0,G_{i}(x^{*})<0\},&\\ &l_{0+}(x^{*})=\{i \vert H_{i}(x^{*})=0,G_{i}(x^{*})>0\},&\\ &~l_{00}(x^{*})=\{i \vert H_{i}(x^{*})=0,G_{i}(x^{*})=0\},&\\ &l_{0-}(x^{*})=\{i \vert H_{i}(x^{*})=0,G_{i}(x^{*})<0\}.& \end{array} $$

We also use the following Lagrangian function and its gradient:

$$ \begin{array}{@{}rcl@{}} L(y, \delta, \mu,\alpha^{H}, \alpha^{G})=f(y)+\sum\limits_{i \in M}\delta_{i}g_{i}(y)+\sum\limits_{i \in P}\mu_{i}h_{i}(y)-\sum\limits_{i \in L}{\alpha_{i}^{H}}H_{i}(y)+\sum\limits_{i \in L}{\alpha_{i}^{G}}G_{i}(y) \end{array} $$

where

$$ \delta := (\delta_{1}, \delta_{2}, {\dots} , \delta_{m}), \mu :=(\mu_{1}, \mu_{2}, \dots, \mu_{p}), \alpha^{H} :=({\alpha_{1}^{H}}, {\alpha_{2}^{H}}, \dots, {\alpha_{l}^{H}}), \alpha^{G}:=({\alpha_{1}^{G}},{\alpha_{2}^{G}}, \dots, {\alpha_{l}^{G}}) $$

and

$$ \begin{array}{@{}rcl@{}} \nabla L(y, \delta, \mu,\alpha^{H}, \alpha^{G})=\nabla f(y)+\sum\limits_{i \in M}\delta_{i}\nabla g_{i}(y)+\sum\limits_{i \in P}\mu_{i} \nabla h_{i}(y)-\sum\limits_{i \in L}{\alpha_{i}^{H}} \nabla H_{i}(y)+\sum\limits_{i \in L}{\alpha_{i}^{G}} \nabla G_{i}(y). \end{array} $$

We define the following index sets for x ∈ X :

$$ \begin{array}{@{}rcl@{}} &{}l_{g}^{+}(x)=\{i \in M \vert \delta_{i} >0\}, &\\ & {\kern-5.5pt}l_{h}^{+}(x)=\{i \in l_{i}(x)\vert \mu_{i} > 0\},& \\ &{} l_{h}^{-}(x)=\{i \in l_{i}(x)\vert \mu_{i} < 0\},& \\ & l_{0+}^{+}(x)=\{i \in l_{0+}(x)\vert {\alpha_{i}^{H}} > 0\},& \\ & {\kern1pt}l_{0+}^{-}(x)=\{i \in l_{0+}(x)\vert {\alpha_{i}^{H}} < 0\},& \\ &{\kern2pt} l_{00}^{+}(x)=\{i \in l_{00}(x)\vert {\alpha_{i}^{H}} > 0\},& \\ &{\kern2pt} l_{+0}^{+}(x)=\{i \in l_{+0}(x)\vert {\alpha_{i}^{H}} > 0\},& \\ & {\kern2pt}l_{+-}^{+}(x)=\{i \in l_{+-}(x)\vert {\alpha_{i}^{H}} > 0\},& \\ &{\kern2pt} l_{0-}^{+}(x)=\{i \in l_{0-}(x)\vert {\alpha_{i}^{H}} > 0\},& \\ & l_{+0}^{++}(x)=\{i \in l_{+0}(x)\vert {\alpha_{i}^{G}} > 0\},& \\ & l_{+-}^{++}(x)=\{i \in l_{+-}(x)\vert {\alpha_{i}^{G}} > 0\}.& \end{array} $$

(2.1)

In order to establish duality results, we give the following definitions and theorem which can be found in Achtziger and Kanzow [1].

Definition 2.3

Let x^∗∈ X be a feasible point of the MPVC. The Abadie constraint qualification, denoted by ACQ, is said to hold at x^∗, iff T(x^∗) = L(x^∗), where

$$T(x^{*})= \left\lbrace d \in \mathbb{R}^{n}: \exists \{x^{k}\} \subseteq X, \exists \{t_{k}\} \downarrow 0, x^{k} \to x^{*} \ \text{and} \ \frac{x^{k}-x^{*}}{t_{k}}\to d \right\rbrace $$

is the standard tangent cone of the MPVC at x^∗, and

$$ \begin{array}{@{}rcl@{}} L(x^{*})=\{d \in \mathbb{R}^{n}: &&\nabla g_{i}(x^{*})^{T}d \leq 0, i \in l_{g}(x^{*}),\\ &&\nabla h_{i}(x^{*})^{T}d=0, i \in P,\\ &&\nabla H_{i}(x^{*})^{T}d=0, i \in l_{0+}(x^{*}),\\ &&\nabla H_{i}(x^{*})^{T}d\geq 0, i \in l_{00}(x^{*}) \cup l_{0-}(x^{*}),\\ &&\nabla G_{i}(x^{*})^{T}d \leq 0, i \in l_{+0}(x^{*}) \} \end{array} $$

denotes the corresponding linearized cone of the MPVC at x^∗.

Definition 2.4

Let x^∗∈ X be a feasible point of the MPVC. The VC-ACQ is said to hold at x^∗, iff $L^{VC}(x^{*}) \subseteq T (x^{*}),$ where

$$ \begin{array}{@{}rcl@{}} L^{VC}(x^{*})=\{d \in \mathbb{R}^{n}: \nabla g_{i}(x^{*})^{T}d &\leq& 0, i \in l_{g}(x^{*}), \\ \nabla h_{i}(x^{*})^{T}d&=&0, i\in P, \\ \nabla H_{i}(x^{*})^{T}d&=&0, i \in l_{0+}(x^{*}), \\ \nabla H_{i}(x^{*})^{T}d&\geq& 0, i \in l_{00}(x^{*}) \cup l_{0-}(x^{*}), \\ \nabla G_{i}(x^{*})^{T}d &\leq& 0, i \in l_{00}(x^{*}) \cup l_{+0}(x^{*}) \} \end{array} $$

denotes the corresponding VC-linearized cone of the MPVC at x^∗.

Theorem 2.1

Let x^∗∈ X be a local minimum of the MPVC such that VC-ACQ holds at x^∗. Then, there exist Lagrange multipliers $\delta _{i} \in \mathbb {R} (i \in M), \mu _{i} \in \mathbb {R}(i \in l_{h}), {\alpha _{i}^{H}}, {\alpha _{i}^{G}} \in \mathbb {R}(i \in L),$ such that

$$ \begin{array}{@{}rcl@{}} \nabla L(x^{*}, \delta, \mu, \alpha^{H}, \alpha^{G}) =0 \end{array} $$

(2.2)

and

$$ \begin{array}{@{}rcl@{}} h_{i}(x^{*})&=&0, \ (i \in l_{h}(x^{*})), \\ \delta_{i} \geq 0, g_{i}(x^{*}) \leq 0, \delta_{i}g_{i}(x^{*})&=&0, \ (i \in M), \\ {\alpha_{i}^{H}}&=&0, \ (i \in l_{+}(x^{*})), \\ {\alpha_{i}^{H}}&\geq& 0, \ (i \in l_{00}(x^{*})\cup l_{0-}(x^{*})), \\ && {\alpha_{i}^{H}} \ \text{is free} \ (i \in l_{0+}(x^{*})), \\ {\alpha_{i}^{G}}&=&0, \ (i \in l_{0+}(x^{*}) \cup l_{0-}(x^{*}) \cup l_{+-}(x^{*})), {\alpha_{i}^{G}}\\ &\geq& 0 (i \in l_{00}(x^{*})\cup l_{+0}(x^{*})). \end{array} $$

(2.3)

The definition of higher order convex function was introduced by Lin and Fukushima [18]. Further, the definitions of higher order invex function and generalized higher order invex functions for nondifferentiable functions was introduced by Joshi [11]. Motivated by the earlier work on higher order invex and generalized invex functions we are introducing the definitions of higher order generalized invex functions. These definitions are the smooth version of the definitions of higher order invex functions and generalized invex functions (Joshi [11]). These concepts of generalized invexity play a vital role during the establishment of duality theorems.

Definition 2.5

Let $S \subseteq \mathbb {R}^{n}$ be any nonempty set and let $f : S \to \mathbb {R}$ be continuously differentiable. Then, f is said to be higher order strongly invex at x^∗∈ S with respect to the kernel function $\xi :S \times S \to \mathbb {R}^{n}$ on S, if for any x ∈ S there exist some c > 0, such that ∀σ > 0, one has

$$ \begin{array}{@{}rcl@{}} f(x)-f(x^{*}) \geq \langle \nabla f(x^{*}), \xi(x,x^{*})\rangle +c \Vert \xi(x,x^{*})\Vert^{\sigma}. \end{array} $$

Definition 2.6

Let $S \subseteq \mathbb {R}^{n}$ be any nonempty set and let $f : S \to \mathbb {R}$ be continuously differentiable. Then, f is said to be higher order strongly pseudoinvex at x^∗∈ S with respect to the kernel function $\xi :S \times S \to \mathbb {R}^{n}$ on S, if for any x ∈ S, there exist some c > 0, such that ∀σ > 0, one has

$$ \begin{array}{@{}rcl@{}} \langle \nabla f(x^{*}), \xi(x,x^{*})\rangle +c \Vert \xi(x,x^{*})\Vert^{\sigma} \geq 0 \implies f(x)\geq f(x^{*}). \end{array} $$

Definition 2.7

Let $S \subseteq \mathbb {R}^{n}$ be any nonempty set and let $f : S \to \mathbb {R}$ be continuously differentiable. Then, f is said to be higher order strongly quasiinvex at x^∗∈ S with respect to the kernel function $\xi :S \times S \to \mathbb {R}^{n}$ on S, if for any x ∈ S, there exist some c > 0, such that ∀σ > 0, one has

$$ \begin{array}{@{}rcl@{}} f(x)\leq f(x^{*}) \implies \langle \nabla f(x^{*}), \xi(x,x^{*})\rangle +c \Vert \xi(x,x^{*})\Vert^{\sigma} \leq 0. \end{array} $$

Now, we give the dual models and establish weak, strong, converse and restricted converse duality theorems. These dual models are taken from [7, 21].

3 Dual models for mathematical programs with vanishing constraints

In this section, we give two dual models, namely Wolfe type dual model and Mond–Weir type dual model respectively.

3.1 Wolfe type dual model

For x ∈ X, the Wolfe type dual of the MPVC , VC-WD(x) for short, is as follows:

$$ \begin{array}{@{}rcl@{}} \max &&L(y, \delta, \mu, \alpha^{H}, \alpha^{G})\\ \text{subject to} && \nabla L (y, \delta, \mu, \alpha^{H}, \alpha^{G}) =0,\\ &&\delta_{i} \geq 0, \forall i \in M,\\ &&{\alpha_{i}^{G}}=\nu_{i}H_{i}(x), \nu_{i} \geq 0, \forall i \in L, \\ &&{\alpha_{i}^{H}}=\rho_{i}-\nu_{i}G_{i}(x), \rho_{i} \geq 0, \forall i \in L. \end{array} $$

(3.1)

Let $S_{W}(x) \subseteq \mathbb {R}^{n} \times \mathbb {R}^{m} \times \mathbb {R}^{p} \times \mathbb {R}^{l} \times \mathbb {R}^{l}$ denote the feasible set, i.e.,

$$ \begin{array}{@{}rcl@{}} S_{W}(x)&=&\{(y, \delta, \mu, \alpha^{H}, \alpha^{G}, \rho, \nu): \nabla L(y,\delta, \mu, \alpha^{H}, \alpha^{G})=0, \\ \delta_{i} &\geq& 0, \forall i \in M,\\ {\alpha_{i}^{G}}&=&\nu_{i}H_{i}(x), \nu_{i} \geq 0, i \in L, \\ {\alpha_{i}^{H}}&=&\rho_{i}-\nu_{i}G_{i}(x), \rho_{i} \geq 0, i \in L \}. \end{array} $$

(3.2)

We denote by

$$prS_{W}(x)= \{y \in \mathbb{R}^{n}: (y,\delta, \mu, \alpha^{H}, \alpha^{G}, \rho, \nu) \in S_{W}(x)\}$$

the projection of the set S_W(x) on $\mathbb {R}^{n}.$ To be independent of the MPVC, the another dual problem which is denoted by VC-WD is as follows:

$$ \begin{array}{@{}rcl@{}} \max {}&&L(y, \delta, \mu, \alpha^{H}, \alpha^{G}) \\ &&s.t. (y, \delta, \mu, \alpha^{H}, \alpha^{G}, \rho, \nu) \in \cap_{x \in X}S_{W}(x). \end{array} $$

(3.3)

The set of all feasible points of the VC-WD is denoted by S_W = ∩_x∈XS_W(x) and the projection of the set S_W on $\mathbb {R}^{n}$ is denoted by prS_W.

Firstly, we give the weak duality theorem. This theorem shows the relationship between a feasible point of the MPVC and a feasible point of the Wolfe type dual.

Theorem 3.1

Let x ∈ X,(y,δ,μ,α^H,α^G,ρ,ν) ∈ S_W be feasible points for the MPVC and the VC-WD, respectively. If one of the following condition holds:

• (1) L(.,δ,μ,α^H,α^G) is higher order strongly invex at y ∈ X ∪ prS_W with respect to the kernel function ξ,

• (2) $f, g_{i}(i \in l_{g}^{+}(x)), h_{i}(i \in l_{h}^{+}(x)),-h_{i}(i \in l_{h}^{-}(x)),-H_{i}(i \in l_{+0}(x)\cup l_{+-}(x) \cup l_{00}(x)\cup l_{0-}(x) \cup l_{0+}^{+}(x)),H_{i} (i \in l_{0+}^{-}(x)), -G_{i}(i \in l_{0+}(x)), G_{i}(i \in l_{00}(x) \cup l_{+0}(x) \cup l_{0-}(x) \cup l_{+-}(x) )$ are higher order strongly invex at y ∈ X ∪ prS_w with respect to the common kernel function ξ;

Then f(x) ≥ L(y,δ,μ,α^H,α^G).

Proof 1

Suppose f(x) < L(y,δ,μ,α^H,α^G), i.e.,

$$ \begin{array}{@{}rcl@{}} f (x) < f (y) + \sum\limits_{i \in M}\delta_{i}g_{i}(y) + \sum\limits_{i \in P}\mu_{i}h_{i}(y) - \sum\limits_{i \in L}{\alpha_{i}^{H}}H_{i}(y) + \sum\limits_{i \in L}{\alpha_{i}^{G}}G_{i}(y). \end{array} $$

(3.4)

Since x ∈ X, then it follows that

$$ \begin{array}{@{}rcl@{}} g_{i}(x)&<& 0, \delta_{i} \geq 0, \ i \notin l_{g}(x), \\ g_{i}(x)&=& 0, \delta_{i} \geq 0, \ i \in l_{g}(x),\\ h_{i}(x)&=& 0, \mu_{i} \in \mathbb{R}, \ i \in l_{h}, \\ -H_{i}(x) &<& 0, {\alpha_{i}^{H}} \geq 0, \ i \in l_{+}(x),\\ -H_{i}(x) &=& 0, {\alpha_{i}^{H}} \in \mathbb{R}, \ i \in l_{0}(x),\\ G_{i}(x) &>& 0, {\alpha_{i}^{G}}=0, \ i \in l_{0+}(x),\\ G_{i}(x) &=& 0, {\alpha_{i}^{G}} \geq 0, \ i \in l_{00}(x) \cup l_{+0}(x), \\ G_{i}(x) &<& 0, {\alpha_{i}^{G}} \geq 0, \ i \in l_{0-}(x) \cup l_{+-}(x), \end{array} $$

that is,

$$ \begin{array}{@{}rcl@{}} \sum\limits_{i \in M}\delta_{i}g_{i}(x)+\sum\limits_{i \in P}\mu_{i}h_{i}(x)-\sum\limits_{i \in L}{\alpha_{i}^{H}}H_{i}(x)+\sum\limits_{i \in L}{\alpha_{i}^{G}}G_{i}(x) \leq 0. \end{array} $$

(3.5)

Adding (3.4) and (3.5), one has

$$ \begin{array}{@{}rcl@{}} &&f (x) +\sum\limits_{i \in M}\delta_{i}g_{i}(x)+\sum\limits_{i \in P}\mu_{i}h_{i}(x)-\sum\limits_{i \in L}{\alpha_{i}^{H}}H_{i}(x)+\sum\limits_{i \in L}{\alpha_{i}^{G}}G_{i}(x) \\ &&~~~~~~~~~~~< f(y)+ \sum\limits_{i \in M}\delta_{i}g_{i}(y)+\sum\limits_{i \in P}\mu_{i}h_{i}(y)-\sum\limits_{i \in L}{\alpha_{i}^{H}}H_{i}(y)+\sum\limits_{i \in L}{\alpha_{i}^{G}}G_{i}(y) \end{array} $$

(3.6)

i.e.,

$$ \begin{array}{@{}rcl@{}} L(x,\delta, \mu, \alpha^{H}, \alpha^{G}) < L(y,\delta, \mu, \alpha^{H}, \alpha^{G}). \end{array} $$

(3.7)

By the higher order strong invexity of L(.,δ,μ,α^H,α^G) with respect to the kernel function ξ, it follows that

$$ L(y,\delta, \mu, \alpha^{H}, \alpha^{G}) +\langle \nabla L(y,\delta, \mu, \alpha^{H}, \alpha^{G}), \xi(x,y) \rangle + c\Vert \xi(x,y) \Vert^{\sigma} \leq L(x,\delta, \mu, \alpha^{H}, \alpha^{G}). $$

In view of the first equation in (3.1), one has

$$ L(x,\delta, \mu, \alpha^{H}, \alpha^{G}) \geq L(y,\delta, \mu, \alpha^{H}, \alpha^{G}).$$

This is a contradiction to (3.7) and hence the result is proved.

(2) By the higher order strong invexity of HCode $g_{i}(i \in l_{g}^{+}(x)), h_{i}(i \in l_{h}^{+}(x)),-h_{i}(i \in l_{h}^{-}(x)),-H_{i}(i \in l_{+0}(x)\cup l_{+-}(x) \cup l_{00}(x)\cup l_{0-}(x)\cup l_{0+}^{+}(x)),H_{i} (i \in l_{0+}^{-}(x)), -G_{i}(i \in l_{0+}(x)), G_{i}(i \in l_{00}(x) \cup l_{+0}(x) \cup l_{0-}(x) \cup l_{+-}(x)),$ with respect to the common kernel function ξ, at y ∈ X ∪ prS_W,x ∈ X, and (y,δ,μ,α^H,α^G,ρ,μ) ∈ S_W, one has

$$ \begin{array}{@{}rcl@{}} g_{i}(y)+ \langle \nabla g_{i}(y), \xi(x,y) \rangle +c_{i} \Vert \xi(x,y)\Vert^{\sigma} &\leq& g_{i}(x) \leq 0, c_{i}>0, \delta_{i} >0, i \in l_{g}^{+}(x), \\ h_{i}(y)+ \langle \nabla h_{i}(y), \xi(x,y) \rangle +c_{i} \Vert \xi(x,y)\Vert^{\sigma} &\leq& h_{i}(x) = 0, c_{i}>0, \mu_{i} >0, j \in l_{h}^{+}(x), \\ h_{i}(y)+ \langle \nabla h_{i}(y), \xi(x,y) \rangle +c_{i} \Vert \xi(x,y)\Vert^{\sigma} &\geq& h_{i}(x) = 0, c_{i}>0, \mu_{i} <0, j \in l_{h}^{-}(x), \\ -H_{i}(y)- \langle \nabla H_{i}(y), \xi(x,y) \rangle -{c_{i}^{H}} \Vert \xi(x,y)\Vert^{\sigma} &\leq& -H_{i}(x) \leq 0, {c_{i}^{H}}>0, {\alpha_{i}^{H}} \geq 0, i \in l_{+0}(x) \\ &&\cup l_{+-}(x) \cup l_{00}(x) \cup l_{0-}(x) \cup l_{0+}^{+}(x), \\ -H_{i}(y)- \langle \nabla H_{i}(y), \xi(x,y) \rangle -{c_{i}^{H}} \Vert \xi(x,y)\Vert^{\sigma} &\leq& -H_{i}(x) = 0, {c_{i}^{H}}\!>\!0, {\alpha_{i}^{H}} < 0, i \in l_{0+}^{-}(x),\\ G_{i}(y)+\langle \nabla G_{i}(y), \xi(x,y) \rangle +{c_{i}^{G}} \Vert \xi(x,y)\Vert^{\sigma} &\geq& G_{i}(x) >0, {c_{i}^{G}}\!>\!0, {\alpha_{i}^{G}}=0, i \in l_{0+}(x),\\ G_{i}(y)+\langle \nabla G_{i}(y), \xi(x,y) \rangle +{c_{i}^{G}} \Vert \xi(x,y)\Vert^{\sigma} &\leq& G_{i}(x) =0, {c_{i}^{G}}\!>\!0, {\alpha_{i}^{G}}\geq 0, i \in l_{+0}(x) \cup l_{00}(x),\\ G_{i}(y)+\!\langle \nabla G_{i}(y), \xi(x,y) \rangle +{c_{i}^{G}} \Vert \xi(x,y)\Vert^{\sigma} &\leq& G_{i}(x) <0, {c_{i}^{G}}\!>\!0, {\alpha_{i}^{G}}\!\geq 0, i \in l_{0-}(x) \cup l_{+-}(x), \end{array} $$

which implies that

$$ \begin{array}{@{}rcl@{}} \sum\limits_{i \in M}\delta_{i}g_{i}(y)&+&\sum\limits_{i \in P}\mu_{i}h_{i}(y)-\sum\limits_{i \in L}{\alpha_{i}^{H}}H_{i}(y)+\sum\limits_{i \in L}{\alpha_{i}^{G}}G_{i}(y) \\ &+& \Bigg \langle \sum\limits_{i \in M}\delta_{i} \nabla g_{i}(y)+\sum\limits_{i \in P}\mu_{i}\nabla h_{i}(y)-\sum\limits_{i \in L}{\alpha_{i}^{H}}\nabla H_{i}(y)+\sum\limits_{i \in L}{\alpha_{i}^{G}}\nabla G_{i}(y), \xi(x,y) \Bigg \rangle\\ &+& c_{i} \Vert \xi(x,y) \Vert^{\sigma}+c_{i} \Vert \xi(x,y) \Vert^{\sigma}-{c_{i}^{H}} \Vert \xi(x,y) \Vert^{\sigma}+{c_{i}^{G}} \Vert \xi(x,y) \Vert^{\sigma} \leq 0. \end{array} $$

(3.8)

Also, by the higher order strong invexity of f at y ∈ X ∪ prS_W, with respect to the kernel function ξ, one has

$$ \begin{array}{@{}rcl@{}} f(y)+ \langle \nabla f(y), \xi(x,y) \rangle +c \Vert \xi(x,y) \Vert^{\sigma} \leq f(x). \end{array} $$

(3.9)

Adding (3.8) and (3.9), one has

$$L(y,\delta, \mu, \alpha^{H}, \alpha^{G}) +\langle \nabla L(y,\delta, \mu, \alpha^{H}, \alpha^{G}), \xi(x,y) \rangle + c \Vert \xi(x,y) \Vert^{\sigma} \leq f(x).$$

In view of the first equation in (3.1), one has

$$L(y,\delta, \mu, \alpha^{H}, \alpha^{G}) \leq f(x)$$

and hence the result is proved. □

The following strong duality theorem gives the condition under which the Wolfe dual is solvable and the global maximum can be obtained.

Theorem 3.2

Let x^∗∈ X be a local minimum of the MPVC, such that the VC-ACQ holds at x^∗. Then, there exist Lagrange multipliers $\bar {\delta } \in \mathbb {R}^{m}, \bar {\mu } \in \mathbb {R}^{p}, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu } \in \mathbb {R}^{l},$ such that $(x^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu })$ is a feasible point of the VC-WD(x^∗) and

$$ \begin{array}{@{}rcl@{}} \sum\limits_{i \in M}\bar{\delta}_{i}g_{i}(x^{*})+\sum\limits_{i \in P}\bar{\mu}_{i}h_{i}(x^{*})-\sum\limits_{i \in L}\bar{\alpha}_{i}^{H}H_{i}(x^{*})+\sum\limits_{i \in L}\bar{\alpha}_{i}^{G}G_{i}(x^{*})=0. \end{array} $$

(3.10)

Moreover, if one of the following condition hold:

• (1) L(.,δ,μ,α^H,α^G) is higher order strongly invex at y ∈ X ∪ prS_W(x^∗) with respect to the kernel function ξ;

• (2) $f, g_{i}(i \in l_{g}^{+}(x^{*})), h_{i}(i \in l_{h}^{+}(x^{*})),-h_{i}(i \in l_{h}^{-}(x^{*})),-H_{i}(i \in l_{+0}(x^{*})\cup l_{+-}(x^{*}) \cup l_{00}(x^{*}) \cup l_{0-}(x^{*})\cup l_{0+}^{+}(x^{*})),H_{i} (i \in l_{0+}^{-}(x^{*})), -G_{i}(i \in l_{0+}(x^{*})), G_{i}(i \in l_{00}(x^{*}) \cup l_{+0}(x^{*}) \cup l_{0-}(x^{*}) \cup l_{+-}(x^{*}) )$ are higher order strongly invex at y ∈ X ∪ prS_W(x^∗) with respect to the kernel function ξ;

Then, $(x^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu })$ is a global maximum of the VC-WD(x^∗), that is,

$$L(x^{*},\bar{\delta}, \bar{\mu}, \bar{\alpha}^{H}, \bar{\alpha}^{G})\geq L (y,\delta, \mu, \alpha^{H}, \alpha^{G}), \forall (y,\delta, \mu, \alpha^{H}, \alpha^{G}) \in S_{W}(x^{*})$$

and

$$f(x^{*})= L(x^{*},\bar{\delta}, \bar{\mu}, \bar{\alpha}^{H}, \bar{\alpha}^{G}).$$

Proof 2

Since x^∗ is local minimum of the MPVC and the VC-ACQ condition is satisfied at x^∗, by Theorem 2.1, it follows that, there exist Lagrange multipliers $\bar {\delta } \in \mathbb {R}^{m}, \bar {\mu } \in \mathbb {R}^{p}, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu } \in \mathbb {R}^{l},$ such that the conditions (2.2) and (2.3) hold and hence $(x^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu })$ is a feasible point of the VC-WD(x^∗). By Theorem 3.1, one has

$$ \begin{array}{@{}rcl@{}} f(x^{*}) \geq L (y,\delta, \mu, \alpha^{H}, \alpha^{G}), \forall (y,\delta, \mu, \alpha^{H}, \alpha^{G}, \rho, \nu) \in S_{W}(x^{*}). \end{array} $$

(3.11)

Adding (3.10) and (3.11), one has

$$L(x^{*},\delta, \mu, \alpha^{H}, \alpha^{G})\geq L (y,\delta, \mu, \alpha^{H}, \alpha^{G}), \forall (y,\delta, \mu, \alpha^{H}, \alpha^{G}, \rho, \nu) \in S_{W}(x^{*})$$

that is, $(x^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu })$ is a global maximum of the VC-WD(x^∗). Also, the local minimum of the MPVC and the global minimum of the VC-WD(x^∗) are equal. □

The following theorem is a converse duality theorem. It gives the condition under which a feasible point of the Wolfe dual generates a global minimum of the MPVC.

Theorem 3.3

Let x ∈ X be any feasible solution of the MPVC and let $(y^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu })$ be a feasible point of the VC-WD such that

$$ \begin{array}{@{}rcl@{}} \bar{\delta}_{i}g_{i}(y^{*}) &\geq& 0, i \in M,\\ \bar{\mu}_{i}h_{i}(y^{*}) &=& 0, i \in P,\\ -\bar{\alpha}_{i}^{H}H_{i}(y^{*}) &\geq& 0, i \in P,\\ \bar{\alpha}_{i}^{G}G_{i}(y^{*}) &\geq& 0, i \in P. \end{array} $$

Moreover, if one of the following condition holds:

• (1) L(.,δ,μ,α^H,α^G) is higher order strongly invex at y^∗∈ X ∪ prS_W with respect to the kernel function ξ;

• (2) $f, g_{i}(i \in l_{g}^{+}(x)), h_{i}(i \in l_{h}^{+}(x)),-h_{i}(i \in l_{h}^{-}(x)),-H_{i}(i \in l_{+0}^{+}(x)\cup l_{+-}^{+}(x) \cup l_{00}^{+}(x)\cup l_{0-}^{+}(x) \cup l_{0+}^{+}(x)),H_{i} (i \in l_{0+}^{-}(x)), G_{i}(i \in l_{+0}^{++}(x) \cup l_{+-}^{++}(x))$ are higher order strongly invex at y^∗∈ X ∪ prS_W with respect to the common kernel function ξ.

Then, y^∗ is a global minimum of the MPVC.

Proof 3

Suppose to the contrary that y^∗ is not a global minimum of the MPVC, i.e., there exists $\tilde {x} \in X$, such that

$$ \begin{array}{@{}rcl@{}} f(\tilde{x})<f(y^{*}). \end{array} $$

(3.12)

(1) Since $\tilde {x}$ and $(y^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu })$ be the feasible point for the MPVC and the VC-WD, respectively. Combining the hypothesis in the theorem, one has

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{i \in M}\bar{\delta}_{i}g_{i}(\tilde{x})+\sum\limits_{i \in P}\bar{\mu}_{i}h_{i}(\tilde{x})-\sum\limits_{i \in L}\bar{\alpha}_{i}^{H}H_{i}(\tilde{x})+\sum\limits_{i \in L}\bar{\alpha}_{i}^{G}G_{i}(\tilde{x}) \leq 0 \leq \sum\limits_{i \in M}\bar{\delta}_{i}g_{i}(y^{*})+ \sum\limits_{i \in P}\bar{\mu}_{i}h_{i}(y^{*}) \\ &&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-\sum\limits_{i \in L}\bar{\alpha}_{i}^{H}H_{i}(y^{*})+\sum\limits_{i \in L}\bar{\alpha}_{i}^{G}G_{i}(y^{*}). \end{array} $$

(3.13)

Adding (3.12) and (3.13), one has

$$L(\tilde{x},\bar{\delta}, \bar{\mu}, \bar{\alpha}^{H}, \bar{\alpha}^{G}) < L(y^{*},\bar{\delta}, \bar{\mu}, \bar{\alpha}^{H}, \bar{\alpha}^{G}).$$

By the higher order strong invexity of L(.,δ,μ,α^H,α^G) with respect to the kernel function ξ, at y^∗∈ X ∪ prS_W; it follows that

$$\langle \nabla L(y^{*},\bar{\delta}, \bar{\mu}, \bar{\alpha}^{H}, \bar{\alpha}^{G}), \xi (\tilde{x},y^{*}) \rangle + c \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} <0, $$

this is a contradiction to the dual constraint of the VC-WD(x) and hence the result is proved. (2) Since $\tilde {x}$ and $(y^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu })$ be the feasible point for the MPVC and the VC-WD, respectively. Combining the hypothesis in the theorem, one has

$$ \begin{array}{@{}rcl@{}} g_{i}(\tilde{x}) &\leq& g_{i}(y^{*}), \ i \in l_{g}^{+}(\tilde{x}), \\ h_{i}(\tilde{x}) &=& h_{i}(y^{*}), \ i \in l_{h}^{+}(\tilde{x}) \cup l_{h}^{-}(\tilde{x}),\\ -H_{i}(\tilde{x}) &\leq& -H_{i}(y^{*}), \ i \in l_{+0}^{+}(\tilde{x}) \cup l_{+-}^{+}(\tilde{x}) \cup l_{00}^{+}(\tilde{x}) \cup l_{0-}^{+}(\tilde{x}) \cup l_{0+}^{+}(\tilde{x}),\\ -H_{i}(\tilde{x}) &\geq& -H_{i}(y^{*}), \ i \in l_{0+}^{-}(\tilde{x}),\\ G_{i}(\tilde{x}) &\leq& G_{i}(y^{*}), \ i \in l_{+0}^{++}(\tilde{x}) \cup l_{+-}^{++}(\tilde{x}). \end{array} $$

By the higher order strong invexity of the function with respect to the common kernel function ξ, it follows that

$$ \begin{array}{@{}rcl@{}} &&\langle \nabla g_{i}(y^{*}), \xi (\tilde{x}, y^{*}) \rangle + c_{i} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} \leq 0, c_{i}>0, \bar{\delta}_{i}>0, \ i \in l_{g}^{+}(\tilde{x}), \\ &&\langle \nabla h_{i}(y^{*}), \xi (\tilde{x}, y^{*}) \rangle + c_{i} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} \leq 0, c_{i} >0, \bar{\mu}_{i}>0, \ i \in l_{h}^{+}(\tilde{x}), \\ &&\langle \nabla h_{i}(y^{*}), \xi (\tilde{x}, y^{*}) \rangle + c_{i} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} \geq 0, c_{i} >0, \bar{\mu}_{i}<0, \ i \in l_{h}^{-}(\tilde{x}), \\ &&-\langle \nabla H_{i}(y^{*}), \xi (\tilde{x}, y^{*}) \rangle - {c_{i}^{H}} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} \leq 0, {c_{i}^{H}}>0, \bar{\alpha}_{i}^{H} \geq 0, \ i \in l_{+0}^{+}(\tilde{x}) \cup l_{+-}^{+}(\tilde{x}) \cup l_{00}^{+}(\tilde{x}) \\ && \cup l_{0-}^{+}(\tilde{x}) \cup l_{0+}^{+}(\tilde{x}), \\ &&-\langle \nabla H_{i}(y^{*}), \xi (\tilde{x}, y^{*}) \rangle- {c_{i}^{H}} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} \geq 0, \bar{\alpha}_{i}^{H} \leq 0, {c_{i}^{H}}>0, \ i \in l_{+0}^{+}(\tilde{x}), \\ &&\langle \nabla G_{i}(y^{*}), \xi (\tilde{x},y^{*}) \rangle + {c_{i}^{G}} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} \leq 0, {c_{i}^{G}}>0, \bar{\alpha}_{i}^{G} \geq 0, \ i \in l_{+0}^{++}(\tilde{x}) \cup l_{+-}^{++}(\tilde{x}), \end{array} $$

which implies that

$$ \begin{array}{@{}rcl@{}} \Bigg\langle \sum\limits_{i \in M}\bar{\delta}_{i}&\nabla g_{i}(y^{*})+\sum\limits_{i \in P}\bar{\mu}_{i}\nabla h_{i}(y^{*})-\sum\limits_{i \in L}\bar{\alpha}_{i}^{H}\nabla H_{i}(y^{*})+\sum\limits_{i \in L}\bar{\alpha}_{i}^{G}\nabla G_{i}(y^{*}), \xi (\tilde{x}, y^{*}) \Bigg \rangle &\\& +c_{i} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma}+c_{i} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma}-{c_{i}^{H}} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma}+{c_{i}^{G}} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} \leq 0.& \end{array} $$

Using the above inequality and (3.1), one has

$$\langle \nabla f(y^{*}), \xi (\tilde{x}, y^{*}) \rangle +c \Vert \xi (\tilde{x}, y^{*}) \Vert^{\sigma} \geq 0. $$

By the higher order strong invexity of f, with respect to the kernel function ξ, it follows that

$$f(\tilde{x}) \geq f(y^{*})$$

this is a contradiction to our hypothesis and hence the result is proved. □

The following theorem is restricted converse duality theorem which gives a sufficient condition for a feasible point of the MPVC to be a global minimum by using the Wolfe dual.

Theorem 3.4

Let x^∗∈ X be any feasible solution of the MPVC and let $(y^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu })$ be a feasible point of the VC-WD such that $f(x^{*})=L(y^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G})$. Moreover, if one of the following conditions hold:

• (1) $L(.,\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G})$ is higher order strongly invex at y^∗∈ X ∪ prS_W with respect to the kernel function ξ,

• (2) $f, g_{i}(i \in l_{g}^{+}(x^{*})), h_{i}(i \in l_{h}^{+}(x^{*})),-h_{i}(i \in l_{h}^{-}(x^{*})),-H_{i}(i \in l_{+0}^{+}(x^{*})\cup l_{+-}^{+}(x^{*}) \cup l_{00}^{+}(x^{*})\cup l_{0-}^{+}(x^{*}) \cup l_{0+}^{+}(x^{*})),H_{i} (i \in l_{0+}^{-}(x^{*})), G_{i}(i \in l_{+0}^{++}(x^{*}) \cup l_{+-}^{++}(x^{*}))$ are higher order strongly invex at y^∗∈ X ∪ prS_W with respect to the common kernel functionξ;

Then, x^∗ is a global minimum of the MPVC.

Proof 4

Suppose to the contrary that x^∗∈ X is not a global minimum of the MPVC, then there exists $\tilde {x} \in X$ such that

$$f(\tilde{x}) < f(x^{*}).$$

Combining the assumption of the theorem, it follows that $f(x^{*})<L(y^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G})$ a contradiction to the Theorem 3.1 and hence the result is proved. □

The following strict converse duality theorem gives a sufficient condition about the uniqueness of a local minimum of the MPVC and a global maximum of the Wolfe dual model.

Theorem 3.5

Let x^∗∈ X be a local minimum for the MPVC such that the VC-ACQ holds at x^∗. Assume the conditions of Theorem 3.2 hold and $(y^{*},\tilde {\delta }, \tilde {\mu }, \tilde {\alpha }^{H}, \tilde {\alpha }^{G}, \tilde {\rho }, \tilde {\nu })$ be a global maximum of the VC-WD(x^∗). If one of the following conditions holds:

• (1) $L(.,\tilde {\delta }, \tilde {\mu }, \tilde {\alpha }^{H}, \tilde {\alpha }^{G})$ is strictly higher order strongly invex at y^∗∈ X ∪ prS_W with respect to the kernel function ξ;

• (2) f is strictly higher order strongly invex and, $g_{i}(i \in l_{g}^{+}(x^{*})), h_{i}(i \in l_{h}^{+}(x^{*})),-h_{i}(i \in l_{h}^{-}(x^{*})), -H_{i}(i \in l_{+0}(x^{*})\cup l_{+-}(x^{*}) \cup l_{00}(x^{*})\cup l_{0-}(x^{*})\cup l_{0+}^{+}(x^{*})),H_{i} (i \in l_{0+}^{-}(x^{*})), -G_{i}(i \in l_{0+}(x^{*})), G_{i}(i \in l_{00}(x^{*}) \cup l_{+0}(x^{*}) \cup l_{0-}(x^{*}) \cup l_{+-}(x^{*}))$ are invex at y^∗∈ X ∪ prS_W with respect to the common kernel function ξ; Then x^∗ = y^∗.

Proof 5

Suppose that x^∗≠y^∗. By Theorem 3.2, there exist Lagrange multipliers $\bar {\delta } \in \mathbb {R}^{m}, \bar {\mu } \in \mathbb {R}^{p}, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu } \in \mathbb {R}^{l},$ such that $(y^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu })$ be a global maximum of the V C − WD(x^∗). Hence,

$$ \begin{array}{@{}rcl@{}} f(x^{*})=L(x^{*},\bar{\delta}, \bar{\mu}, \bar{\alpha}^{H}, \bar{\alpha}^{G})=L(y^{*},\tilde{\delta}, \tilde{\mu}, \tilde{\alpha}^{H}, \tilde{\alpha}^{G}). \end{array} $$

(3.14)

In view of the feasibility of x^∗ for the MPVC and the feasibility of $(y^{*},\tilde {\delta }, \tilde {\mu }, \tilde {\alpha }^{H}, \tilde {\alpha }^{G}, \tilde {\rho }, \tilde {\nu })$ for the VC-WD(x^∗), it follows that

$$ \begin{array}{@{}rcl@{}} &g_{i}(x^{*})<0, \tilde{\delta}_{i} \geq 0, \ i \notin l_{g}(x^{*}), & \\ &g_{i}(x^{*})=0, \tilde{\delta}_{i} \geq 0, \ i \in l_{g}(x^{*}), &\\ &h_{i}(x^{*})=0, \tilde{\mu}_{i} \in \mathbb{R}, \ i \in l_{h}(x^{*}),& \\ &-H_{i}(x^{*}) <0, \tilde{\alpha}_{i}^{H} \geq 0, i \in l_{+}(x^{*}),&\\ &-H_{i}(x^{*}) =0, \tilde{\alpha}_{i}^{H} \in \mathbb{R}, \ i \in l_{0}(x^{*}), &\\ &G_{i}(x^{*}) > 0, \tilde{\alpha}_{i}^{G}=0, \ i \in l_{0+}(x^{*}), &\\ &G_{i}(x^{*}) = 0, \tilde{\alpha}_{i}^{G} \geq 0, \ i \in l_{00}(x^{*}) \cup l_{+0}(x^{*}), &\\ &G_{i}(x^{*}) < 0, \tilde{\alpha}_{i}^{G} \geq 0, \ i \in l_{0-}(x^{*}) \cup l_{+-}(x^{*}), & \end{array} $$

that is,

$$ \begin{array}{@{}rcl@{}} \sum\limits_{i \in M}\tilde{\delta}_{i}g_{i}(x^{*})+\sum\limits_{i \in P}\tilde{\mu}_{i}h_{i}(x^{*})-\sum\limits_{i \in L}\tilde{\alpha}_{i}^{H}H_{i}(x^{*})+\sum\limits_{i \in L} \tilde{\alpha}_{i}^{G}G_{i}(x^{*}) \leq 0. \end{array} $$

(3.15)

Adding (3.14) and (3.15), one has

$$ \begin{array}{@{}rcl@{}} L(x^{*},\tilde{\delta}, \tilde{\mu}, \tilde{\alpha}^{H}, \tilde{\alpha}^{G}) \leq L(y^{*},\tilde{\delta}, \tilde{\mu}, \tilde{\alpha}^{H}, \tilde{\alpha}^{G}). \end{array} $$

(3.16)

By the strict higher order strong invexity of $L(.,\tilde {\delta }, \tilde {\mu }, \tilde {\alpha }^{H}, \tilde {\alpha }^{G})$, with respect to the common kernel function ξ, it follows that

$$ \langle \nabla L(y^{*},\tilde{\delta}, \tilde{\mu}, \tilde{\alpha}^{H}, \tilde{\alpha}^{G}), \xi(x^{*},y^{*}) \rangle + c \Vert \xi(x^{*},y^{*}) \Vert^{\sigma} < 0.$$

That is a contradiction to the first equation in (3.1) and hence the result is proved. (2) By the strict higher order strong invexity of f at y^∗, with respect to the kernel function ξ, one has

$$ \begin{array}{@{}rcl@{}} f(x^{*})-f(y^{*}) > \langle \nabla f(y^{*}), \xi(x^{*}, y^{*}) \rangle+ c \Vert \xi(x^{*},y^{*}) \Vert^{\sigma}. \end{array} $$

(3.17)

By the higher order strong invexity of $g_{i}(i \in l_{g}^{+}(x^{*})), h_{i}(i \in l_{h}^{+}(x^{*})),-h_{i}(i \in l_{h}^{-}(x^{*})),-H_{i}(i \in l_{+0}(x^{*})\cup l_{+-}(x^{*})\cup l_{00}(x^{*}) \cup l_{0-}(x^{*})\cup l_{0+}^{+}(x^{*})),H_{i} (i \in l_{0+}^{-}(x^{*})), -G_{i}(i \in l_{0+}(x^{*})), G_{i}(i \in l_{00}(x^{*}) \cup l_{+0}(x^{*}) \cup l_{0-}(x^{*}) \cup l_{+-}(x^{*})),$ at y^∗∈ X ∪ prS_W(x^∗),x^∗∈ X, with respect to the common kernel function ξ and $(y^{*},\tilde {\delta }, \tilde {\mu }, \tilde {\alpha }^{H}, \tilde {\alpha }^{G}, \tilde {\rho }, \tilde {\nu }) \in S_{W}(x^{*}),$ one has

$$ \begin{array}{@{}rcl@{}} &g_{i}(y^{*})+ \langle \nabla g_{i}(y^{*}), \xi (x^{*},y^{*}) \rangle + c_{i} \Vert \xi(x^{*},y^{*}) \Vert^{\sigma} \leq g_{i}(x^{*}) \leq 0, c_{i}>0, \tilde{\delta}_{i} >0, i \in l_{g}^{+}(x^{*}),& \\ &h_{i}(y^{*})+ \langle \nabla h_{i}(y^{*}), \xi (x^{*},y^{*}) \rangle + c_{i} \Vert \xi(x^{*},y^{*})\Vert^{\sigma} \leq h_{i}(x^{*}) = 0, c_{i}>0, \tilde{\mu}_{i} >0, j \in l_{h}^{+}(x^{*}),& \\ &h_{i}(y^{*})+ \langle \nabla h_{i}(y^{*}), \xi (x^{*},y^{*}) \rangle + c_{i} \Vert \xi(x^{*},y^{*})\Vert^{\sigma} \geq h_{i}(x^{*}) = 0, c_{i}>0, \tilde{\mu}_{i} <0, j \in l_{h}^{-}(x^{*}), &\\ &-H_{i}(y^{*})- \langle \nabla H_{i}(y^{*}), \xi (x^{*},y^{*}) \rangle - {c_{i}^{H}} \Vert \xi(x^{*},y^{*})\Vert^{\sigma} \leq -H_{i}(x^{*}) \leq 0, , {c_{i}^{H}}>0, \tilde{\alpha}_{i}^{H} \geq 0, &\\ & i \in l_{+0}(x^{*}) \cup l_{+-}(x^{*}) \cup l_{00}(x^{*}) \cup l_{0-}(x^{*}) \cup l_{0+}^{+}(x^{*}), &\\ &-H_{i}(y^{*})- \langle \nabla H_{i}(y^{*}), \xi (x^{*},y^{*}) \rangle- {c_{i}^{H}} \Vert \xi(x^{*},y^{*})\Vert^{\sigma} \leq -H_{i}(x^{*}) = 0, {c_{i}^{H}}>0, \tilde{\alpha}_{i}^{H} < 0, & \\ & i \in l_{0+}^{-}(x^{*}),& \\ &G_{i}(y^{*})+\langle \nabla G_{i}(y^{*}), \xi (x^{*},y^{*}) \rangle + {c_{i}^{G}} \Vert \xi(x^{*},y^{*})\Vert^{\sigma} \geq G_{i}(x^{*}) >0, {c_{i}^{G}}>0, \tilde{\alpha}_{i}^{G}=0, & \\ & i \in l_{0+}(x^{*}),&\\ &G_{i}(y^{*})+\langle \nabla G_{i}(y^{*}), \xi (x^{*},y^{*}) \rangle + {c_{i}^{G}} \Vert \xi(x^{*},y^{*})\Vert^{\sigma} \leq G_{i}(x^{*}) =0, {c_{i}^{G}}>0, \tilde{\alpha}_{i}^{G}\geq 0, i \in l_{+0}(x^{*}) & \\ & \cup l_{00}(x^{*}),&\\ &G_{i}(y^{*})+\langle \nabla G_{i}(y^{*}), \xi (x^{*},y^{*}) \rangle + {c_{i}^{G}} \Vert \xi(x^{*},y^{*})\Vert^{\sigma} \leq G_{i}(x^{*}) <0, {c_{i}^{G}}>0, \tilde{\alpha}_{i}^{G}\geq 0, i \in l_{0-}(x^{*}) &\\ & \cup l_{+-}(x^{*}),& \end{array} $$

this implies that

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{i \in M}\tilde{\delta}_{i}g_{i}(y^{*})+\sum\limits_{i \in P}\tilde{\mu}_{i}h_{i}(y^{*})-\sum\limits_{i \in L}\tilde{\alpha}_{i}^{H}H_{i}(y^{*})+\sum\limits_{i \in L}\tilde{\alpha}_{i}^{G}G_{i}(y^{*}) \\ &&+ \Bigg \langle \sum\limits_{i \in M}\tilde{\delta}_{i} \nabla g_{i}(y^{*})+\sum\limits_{i \in P}\tilde{\mu}_{i}\nabla h_{i}(y^{*})-\sum\limits_{i \in L}\tilde{\alpha}_{i}^{H}\nabla H_{i}(y^{*})+\sum\limits_{i \in L}\tilde{\alpha}_{i}^{G}\nabla G_{i}(y^{*}), \xi(x^{*},y^{*}) \Bigg \rangle\\ &&+ c_{i} \Vert \xi(x^{*},y^{*})\Vert^{\sigma} + c_{i} \Vert \xi(x^{*},y^{*})\Vert^{\sigma} - {c_{i}^{H}} \Vert \xi(x^{*},y^{*})\Vert^{\sigma} + {c_{i}^{G}} \Vert \xi(x^{*},y^{*})\Vert^{\sigma} \!\leq\! 0. \end{array} $$

(3.18)

Adding (3.17) and (3.18), one has

$$L(y^{*},\tilde{\delta}, \tilde{\mu}, \tilde{\alpha}^{H}, \tilde{\alpha}^{G})< f(x^{*}).$$

This is a contradiction to (3.14) and hence the result is proved. □

In order to verify the theorems, we give the following example.

Example 3.1

Consider the optimization problem

$$ \begin{array}{@{}rcl@{}} \min &&f(x) = {x_{1}^{2}}-{x_{2}^{2}}\\ &&s.t. H_{1}(x)= x_{1}-{x_{2}^{2}} \geq 0,\\ &&G_{1}(x)H_{1}(x)=(x_{1}+{x_{2}^{2}})(x_{1}-{x_{2}^{2}})\leq 0, \end{array} $$

(3.19)

with n = 2,m = p = 0,l = 1. Here, the set of all feasible points X is given by

$$X:=\left\lbrace (x_{1},x_{2}): x_{1}-{x_{2}^{2}} \geq 0, (x_{1}+{x_{2}^{2}})(x_{1}-{x_{2}^{2}}) \leq 0 \right\rbrace.$$

For any feasible x ∈ X, the Wolfe dual model to (3.19) is given by

$$ \begin{array}{@{}rcl@{}} \max &&L(y, {\alpha_{1}^{H}}, {\alpha_{1}^{G}})={y_{1}^{2}}-{y_{2}^{2}}-{\alpha_{1}^{H}}(y_{1}-{y_{2}^{2}})+{\alpha_{1}^{G}}(y_{1}+{y_{2}^{2}})\\ &&s.t. \nabla L(y, {\alpha_{1}^{H}}, {\alpha_{1}^{G}})=\left( 2y_{1} - {\alpha_{1}^{H}}+{\alpha_{1}^{G}}, -2y_{2}+2y_{2}{\alpha_{1}^{H}}+2y_{2}{\alpha_{1}^{G}} \right)=(0,0),\\ && {\alpha_{1}^{G}} =\nu_{1}(x_{1}-{x_{2}^{2}}), \nu_{1} \geq 0,\\ &&{\alpha_{1}^{H}}=\rho_{1}-\nu_{1}(x_{1}+{x_{2}^{2}}), \rho_{1} \geq 0. \end{array} $$

(3.20)

• (1) To show that feasible point x^∗ = (0,0) ∈ X is a global minimum of the MPVC, we have to show that $f(x^{*})=L(y^{*}, \bar {\alpha }_{1}^{H}, \bar {\alpha }_{1}^{G})$ for some $(y^{*}, \bar {\alpha }_{1}^{H}, \bar {\alpha }_{1}^{G}) \in S_{W}$ such that the hypothesis of Theorem 3.4, holds at y^∗ on $X \cup pr_{\mathbb {R}^{2}}S_{W}$.

  The feasible set S_W of the VC-WD is given by

  $$ \begin{array}{@{}rcl@{}} S_{W}:=\{(y_{1},y_{2},{\alpha_{1}^{H}}, {\alpha_{1}^{G}}, \rho_{1}, \kappa_{1}): 2y_{1}- {\alpha_{1}^{H}}+{\alpha_{1}^{G}}=0,\\ &&-2y_{2}+2y_{2}{\alpha_{1}^{H}}+2y_{2}{\alpha_{1}^{G}}=0,\\ &&{\alpha_{1}^{G}}=\nu_{1}H_{1}(x), \nu_{1} \geq 0,\\ &&{\alpha_{1}^{H}}=\rho_{1}-\nu_{1}G_{1}(x), \rho_{1} \geq 0\}. \end{array} $$

  Also from (3.19), we can get (0,0) as a feasible point and from (3.20), we have ${\alpha _{1}^{G}}=0, {\alpha _{1}^{H}} \geq 0$ and for ${\alpha _{1}^{H}} + {\alpha _{1}^{G}} \neq 1$, we obtain

  $$L(y_{1},y_{2}, {\alpha_{1}^{H}}, {\alpha_{1}^{G}})=-\frac{1}{2}({\alpha_{1}^{H}})^{2}-\frac{1}{2}({\alpha_{1}^{G}})^{2}-\frac{1}{2}({\alpha_{1}^{H}}{\alpha_{1}^{G}}) \leq 0, $$

  and it is very easy to see that $f(x_{1},x_{2})=L(y_{1},y_{2}, {\alpha _{1}^{H}}, {\alpha _{1}^{G}})$ is possible only if x^∗ = (0,0). That is x^∗ = (0,0) ∈ X, and $(y, {\alpha _{1}^{G}}, {\alpha _{1}^{H}}, \rho _{1}, \nu _{1})=(0,0,0,0,0) \in S_{W}(x^{*}),$ one has

  $$f(x^{*})=0=L(0,0,0).$$

  It can be verified that the hypothesis of Theorem 3.4 holds. Taking account (3.19), x^∗ is a global minimum of (3.19). So, Theorem 3.4 is verified.

• (2) From (3.20), we can get $y_{1}=\frac {{\alpha _{1}^{H}}-{\alpha _{1}^{G}}}{2},$ and for ${\alpha _{1}^{H}}+{\alpha _{1}^{G}} \neq 1,$ we have y₂ = 0. One also has

  $$ L(y,{\alpha_{1}^{H}}, {\alpha_{1}^{G}})= -\frac{1}{2}({\alpha_{1}^{H}})^{2}-\frac{1}{2}({\alpha_{1}^{G}})^{2}-\frac{1}{2}({\alpha_{1}^{H}}{\alpha_{1}^{G}}) \leq 0. $$

  Since x^∗ = (0,0) is the solution of the MPVC and VC-ACQ holds at x^∗. We can get $f(x) \geq L(y, {\alpha _{1}^{H}}, {\alpha _{1}^{G}}).$ Hence, Theorem 3.1 is verified.

• (3) One can see that (3.19) satisfy VC-LICQ. So we obtain (3.19) satisfies VC-ACQ. By Theorem 2.1, there exist Lagrange multipliers ${\alpha _{1}^{H}}, {\alpha _{1}^{G}}, \rho _{1}, \nu _{1} \in \mathbb {R}$ such that $(0, {\alpha _{1}^{H}}, {\alpha _{1}^{G}}, \rho _{1}, \nu _{1})$ is a feasible point of the VC-WD(0) and

  $$ -{\alpha_{1}^{H}}H_{1}(0)+ {\alpha_{1}^{G}}G_{1}(0)=0. $$

  So, $(0, {\alpha _{1}^{H}}, {\alpha _{1}^{G}}, \rho _{1}, \nu _{1})$ is a global maximum of the VC-WD(0) and $f(0) = 0 = L(0, {\alpha _{1}^{H}}, {\alpha _{1}^{G}}).$ Theorem 3.2 is verified.

3.2 Mond–Weir type dual model

In this section, we discuss Mond–Weir type dual for MPVC. For x ∈ X, the Mond–Weir type dual of the MPVC, VC-MWD(x) for short, is as follows:

$$ \begin{array}{@{}rcl@{}} \max && f(y)\\ &&s.t. \nabla L(y, \delta, \mu, \alpha^{H}, \alpha^{G}) =0,\\ &&\delta_{i} \geq 0, \delta_{i}g_{i}(y) \geq 0, i \in M, \\ &&\mu_{i}h_{i}(y) =0, i \in P,\\ &&{\alpha_{i}^{G}}G_{i}(y) \geq 0, i \in L,\\ &&{\alpha_{i}^{G}}= \nu_{i}H_{i}(x), \nu_{i} \geq 0, i \in L,\\ &&-{\alpha_{i}^{H}}H_{i}(y) \geq 0, i \in L,\\ &&{\alpha_{i}^{H}}=\rho_{i}-\nu_{i}G_{i}(x), \rho_{i} \geq 0, i \in L. \end{array} $$

(3.21)

Let $S_{MW}(x)\subseteq \mathbb {R}^{n} \times \mathbb {R}^{m} \times \mathbb {R}^{p} \times \mathbb {R}^{l} \times \mathbb {R}^{l}$ denote feasible point set, i.e.,

$$ \begin{array}{@{}rcl@{}} S_{MW}(x)&=&\{(y, \delta, \mu, \alpha^{H}, \alpha^{G},\rho, \nu): \nabla L(y, \delta, \mu, \alpha^{H}, \alpha^{G})=0,\\ \delta_{i} &\geq& 0, \delta_{i}g_{i}(y) \geq 0, i \in M,\\ \mu_{i}h_{i}(y) &=&0, i \in P,\\ {\alpha_{i}^{G}}G_{i}(y) &\geq& 0, i \in L,\\ {\alpha_{i}^{G}}&=& \nu_{i}H_{i}(x), \nu_{i} \geq 0, i \in L,\\ &&-{\alpha_{i}^{H}}H_{i}(y) \geq 0, i \in L,\\ {\alpha_{i}^{H}}&=&\rho_{i}-\nu_{i}G_{i}(x), \rho_{i} \geq 0, i \in L \}. \end{array} $$

(3.22)

We denote by

$$prS_{MW}(x)=\{y \in \mathbb{R}^{n}: (y, \delta, \mu, \alpha^{H}, \alpha^{G},\rho, \nu) \in S_{MW}(x)\}$$

the projection of the set S_MW(x) on $\mathbb {R}^{n}.$

Similar to the Wolfe dual, we also consider another dual problem which is denoted by VC-MWD as follows:

$$ \begin{array}{@{}rcl@{}} \max &f (y) &\\ & s.t. (y, \delta, \mu, \alpha^{H}, \alpha^{G},\rho, \nu) \in \cap_{x \in X}S_{MW}(x). \end{array} $$

The set of all feasible points of the VC-MWD is denoted by S_MW = ∩_x∈XS_MW(x) and the projection of the set S_MW on $\mathbb {R}^{n}$ is denoted by prS_MW.

The following weak duality theorem shows the relationship between a feasible point of the MPVC and a feasible point of the Mond–Weir type dual.

Theorem 3.6

Let x ∈ X and (y,δ,μ,α^H,α^G,ρ,ν) ∈ S_MW be feasible points for the MPVC and the VC-MWD, respectively. Moreover, if one of the following condition holds:

• (1) f(.) is higher order strongly pseudoinvex and $\sum \limits _{i \in M}\delta _{i}g_{i}(.)+\sum \limits _{i \in P}\mu _{i}h_{i}(.)-\sum \limits _{i \in L}{\alpha _{i}^{H}}H_{i}(.)+ \sum \limits _{i \in L}{\alpha _{i}^{G}}G_{i}(.)$ is quasiinvex at y ∈ X ∪ prS_MW, with respect to the common kernel function ξ;

• (2) f(.) is higher order strongly pseudoinvex and $g_{i}(i \in l_{g}^{+}(x)), h_{i}(i \in l_{h}^{+}(x)),-h_{i}(i \in l_{h}^{-}(x)),-H_{i}(i \in l_{+0}^{+}(x)\cup l_{+-}^{+}(x)\cup l_{00}^{+}(x)\cup l_{0-}^{+}(x)\cup l_{0+}^{+}(x)),H_{i} (i \in l_{0+}^{-}(x)), G_{i}(i \in l_{+0}^{++}(x)\cup l_{+-}^{++}(x) )$ are higher order strongly quasiinvex at y ∈ X ∪ prS_MW with respect to the common kernel function ξ;

Then, f(x) ≥ f(y).

Proof 6

Since x ∈ X and (y,δ,μ,α^H,α^G,ρ,ν) ∈ S_MW, it follows that

$$ \begin{array}{@{}rcl@{}} &&g_{i}(x) \leq 0, \ \delta_{i} \geq 0, i \in M,\\ &&h_{i}(x)=0, \mu_{i} \in \mathbb{R}, i \in l_{h},\\ &&-H_{i}(x) <0, {\alpha_{i}^{H}} \geq 0, i \in l_{+}(x),\\ &&-H_{i}(x) =0, {\alpha_{i}^{H}} \in \mathbb{R}, i \in l_{0}(x),\\ &&G_{i}(x) > 0, {\alpha_{i}^{G}}=0, i \in l_{0+}(x),\\ &&G_{i}(x) = 0, {\alpha_{i}^{G}} \geq 0, i \in l_{00}(x) \cup l_{+0}(x),\\ &&G_{i}(x) < 0, {\alpha_{i}^{G}} \geq 0, i \in l_{0-}(x) \cup l_{+-}(x). \end{array} $$

By (3.21), it implies that

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{i \in M}\delta_{i}g_{i}(x)+\sum\limits_{i \in P}\mu_{i}h_{i}(x)-\sum\limits_{i \in L}{\alpha_{i}^{H}}H_{i}(x)+\sum\limits_{i \in L}{\alpha_{i}^{G}}G_{i}(x)\\ &\leq& \sum\limits_{i \in M}\delta_{i}g_{i}(y)+\sum\limits_{i \in P}\mu_{i}h_{i}(y)-\sum\limits_{i \in L}{\alpha_{i}^{H}}H_{i}(y)+\sum\limits_{i \in L}{\alpha_{i}^{G}}G_{i}(y). \end{array} $$

Combining the higher order strong quasiinvexity of $\sum \limits _{i \in M}\delta _{i}g_{i}(.)+\sum \limits _{i \in P}\mu _{i}h_{i}(.)-\sum \limits _{i \in L}{\alpha _{i}^{H}}H_{i}(.)+\sum \limits _{i \in L}{\alpha _{i}^{G}}G_{i}(.)$ with respect to the common kernel function ξ, one has

$$ \begin{array}{@{}rcl@{}} &&\Bigg\langle \sum\limits_{i \in M}\delta_{i}\nabla g_{i}(y)+\sum\limits_{i \in P}\mu_{i} \nabla h_{i}(y)-\sum\limits_{i \in L}{\alpha_{i}^{H}} \nabla H_{i}(y)+\sum\limits_{i \in L}{\alpha_{i}^{G}} \nabla G_{i}(y), \xi(x,y)\Bigg \rangle\\ &&+c_{i} \Vert \xi(x,y) \Vert^{\sigma}+c_{i} \Vert \xi(x,y) \Vert^{\sigma} - {c_{i}^{H}} \Vert \xi(x,y) \Vert^{\sigma} + {c_{i}^{G}} \Vert \xi(x,y) \Vert^{\sigma} \leq 0. \end{array} $$

Using the above inequality and the first equation in (3.21), one has

$$\langle \nabla f(y), \xi(x,y) \rangle +c \Vert \xi(x,y) \Vert^{\sigma} \geq 0.$$

By the higher order strong pseudoinvexity of f with respect to the kernel function ξ it implies that

$$f(x) \geq f(y)$$

and hence the result is proved. (2) By x ∈ X, and (y,δ,μ,α^H,α^G,ρ,ν) ∈ S_MW it follows that

$$ \begin{array}{@{}rcl@{}} &&g_{i}(x) \leq g_{i}(y), \ i \in l_{g}^{+}(x),\\ &&h_{i}(x) = h_{i}(y), \ i \in l_{h}^{+}(x) \cup l_{h}^{-}(x),\\ &&-H_{i}(x) \leq -H_{i}(y), \ i \in l_{+0}^{+}(x) \cup l_{+-}^{+}(x) \cup l_{00}^{+}(x) \cup l_{0-}^{+}(x) \cup l_{0+}^{+}(x),\\ &&-H_{i}(x) \geq -H_{i}(y), \ i \in l_{0+}^{-}(x),\\ &&G_{i}(x) \leq G_{i}(y), \ i \in l_{+0}^{++}(x) \cup l_{+-}^{++}(x). \end{array} $$

By the higher order strong quasiinvexity of $g_{i}(i \in l_{g}^{+}(x)), h_{i}(i \in l_{h}^{+}(x)),-h_{i}(i \in l_{h}^{-}(x)), -H_{i}(i \in l_{+0}^{+}(x)\cup l_{+-}^{+}(x)\cup l_{00}^{+}(x)\cup l_{0-}^{+}(x)\cup l_{0+}^{+}(x)),H_{i} (i \in l_{0+}^{-}(x)), G_{i}(i \in l_{+0}^{++}(x)\cup l_{+-}^{++}(x) )$ with respect to the common kernel function ξ, it follows that

$$ \begin{array}{@{}rcl@{}} &\langle \nabla g_{i}(y), \xi(x,y) \rangle +c_{i} \Vert \xi(x,y) \Vert^{\sigma} \leq 0, c_{i}>0, {\delta}_{i}>0, \ i \in l_{g}^{+}(x), &\\ &\langle \nabla h_{i}(y), \xi(x,y) \rangle +c_{i} \Vert \xi(x,y) \Vert^{\sigma} \leq 0, c_{i}>0, {\mu}_{i}>0, \ i \in l_{h}^{+}(x), &\\ &\langle \nabla h_{i}(y), \xi(x,y) \rangle +c_{i} \Vert \xi(x,y) \Vert^{\sigma} \geq 0, c_{i}>0, {\mu}_{i}<0, \ i \in l_{h}^{-}(x), &\\ &-\langle \nabla H_{i}(y), \xi(x,y) \rangle -{c_{i}^{H}} \Vert \xi(x,y) \Vert^{\sigma} \leq 0, {c_{i}^{H}}>0, {\alpha}_{i}^{H}, \geq 0, \ i \in l_{+0}^{+}(x) \cup l_{+-}^{+}(x) \cup l_{00}^{+}(x) \cup l_{0-}^{+}(x)&\\ & \cup l_{0+}^{+}(x), &\\ &-\langle \nabla H_{i}(y), \xi(x,y) \rangle -{c_{i}^{H}} \Vert \xi(x,y) \Vert^{\sigma} \geq 0, {c_{i}^{H}}>0, {\alpha}_{i}^{H} \leq 0, \ i \in l_{+0}^{-}(x), &\\ &\langle \nabla G_{i}(y), \xi(x,y) \rangle +{c_{i}^{G}} \Vert \xi(x,y) \Vert^{\sigma} \leq 0, {c_{i}^{G}}>0, {\alpha}_{i}^{G} \geq 0, \ i \in l_{+0}^{++}(x) \cup l_{+-}^{++}(x).& \end{array} $$

From the above inequalities and (2.1), it follows that

$$ \begin{array}{@{}rcl@{}} &&\Bigg\langle \sum\limits_{i \in M}\delta_{i}\nabla g_{i} (y)+\sum\limits_{i \in P}\mu_{i} \nabla h_{i}(y)-\sum\limits_{i \in L}{\alpha_{i}^{H}} \nabla H_{i}(y)+\sum\limits_{i \in L}{\alpha_{i}^{G}} \nabla G_{i}(y), \xi(x,y) \Bigg\rangle\\ &&+c_{i} \Vert \xi(x,y) \Vert^{\sigma}+c_{i} \Vert \xi(x,y) \Vert^{\sigma} - {c_{i}^{H}} \Vert \xi(x,y) \Vert^{\sigma}+{c_{i}^{G}} \Vert \xi(x,y) \Vert^{\sigma} \!\leq\! 0. \end{array} $$

Combining the above inequality and (3.21), one has

$$\langle \nabla f(y), \xi(x,y) \rangle +c \Vert \xi(x,y) \Vert^{\sigma} \geq 0.$$

By the higher order strong pseudoinvexity of f, with respect to the kernel function ξ it implies that

$$f (x) \geq f(y)$$

and hence the result is proved. □

The following strong duality theorem gives the condition under which the Mond–Weir dual is solvable and the global maximum can be obtained.

Theorem 3.7

Let x^∗∈ X be a local minimum of the MPVC, such that the VC-ACQ holds at x^∗. Then, there exist Lagrange multipliers $\bar {\delta } \in \mathbb {R}^{m}, \bar {\mu } \in \mathbb {R}^{p}, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu } \in \mathbb {R}^{l},$ such that $(x^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu })$ is a feasible point of the VC-MWD(x^∗), that is, $(x^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu }) \in S_{MW}(x^{*})$. Moreover, Theorem 3.6 holds, then $(x^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G})$ is a global maximum of the VC-MWD(x^∗).

Proof 7

Since x^∗∈ X is a local minimum of the MPVC and the VC-ACQ condition is satisfied at x^∗. By Theorem 2.1, it follows that, there exist Lagrange multipliers $\bar {\delta } \in \mathbb {R}^{m}, \bar {\mu } \in \mathbb {R}^{p}, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu } \in \mathbb {R}^{l},$ such that the conditions (2.2) and (2.3) hold and hence $(x^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu })$ is a feasible point of VC-MWD(x^∗). By Theorem 3.6, it follows that

$$f(x^{*}) \geq f(y), \forall (y,{\delta}, {\mu}, {\alpha}^{H}, {\alpha}^{G}, {\rho}, {\nu}) \in S_{MW}(x^{*})$$

and hence $(x^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu }) \in S_{MW}(x^{*})$ is a global maximum of the VC-MWD. □

The following converse duality theorem gives the condition under which a feasible point of the Mond–Weir dual generates a global minimum of the MPVC.

Theorem 3.8

Let x ∈ X and $(y^{*}, \bar {\delta },\bar { \mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G},\bar {\rho }, \bar {\nu }) \in S_{MW}$ be feasible points for the MPVC and the VC-MWD, respectively. Moreover, if one of the following condition holds:

• (1) f(.) is higher order strongly pseudoinvex and $\sum \limits _{i \in M}\bar {\delta }_{i}g_{i}(.)+\sum \limits _{i \in P}\bar {\mu }_{i}h_{i}(.)-\sum \limits _{i \in L}\bar {\alpha }_{i}^{H}H_{i}(.)+ \sum \limits _{i \in L}\bar {\alpha }_{i}^{G}G_{i}(.)$ is quasiinvex at y ∈ X ∪ prS_MW, with respect to the common kernel function ξ;

• (2) f is higher order strongly pseudoinvex and $g_{i}(i \in l_{g}^{+}(x)), h_{i}(i \in l_{h}^{+}(x)),-h_{i}(i \in l_{h}^{-}(x)), -H_{i}(i \in l_{+0}^{+}(x)\cup l_{+-}^{+}(x) \cup l_{00}^{+}(x)\cup l_{0-}^{+}(x)\cup l_{0+}^{+}(x)),H_{i} (i \in l_{0+}^{-}(x)), G_{i}(i \in l_{+0}^{++}(x) \cup l_{+-}^{++}(x) )$ are higher order strongly quasiinvex at y ∈ X ∪ prS_MW with respect to the common kernel function ξ;

Then y^∗ is a global minimum of the MPVC.

Proof 8

Suppose to the contrary that y^∗ is not a global minimum of the MPVC, that is, there exists $\tilde {x} \in X,$ such that $f(\tilde {x}) < f(y^{*}).$

• (1) By the higher order strong pseudoinvexity of f(.), with respect to the kernel function ξ, one has

  $$ \begin{array}{@{}rcl@{}} \langle \nabla f(y^{*}), \xi (\tilde{x},y^{*}) \rangle +c \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} <0. \end{array} $$

  (3.23)

  Since $\tilde {x} \in X$ and $(y^{*}, \bar {\delta },\bar { \mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G},\bar {\rho }, \bar {\nu }) \in S_{MW}$, one has

  $$ \begin{array}{@{}rcl@{}} &\bar{\delta}_{i}g_{i}(\tilde{x}) \leq \bar{\delta}_{i}g_{i}(y^{*}), \ i \in M,&\\ &\bar{\mu}_{i}h_{i}(\tilde{x}) = \bar{\mu}_{i}h_{i}(y^{*}), \ i \in P, &\\ &-\bar{\alpha}_{i}^{H}H_{i}(\tilde{x}) \leq -\bar{\alpha}_{i}^{H}H_{i}(y^{*}), \ i \in L,&\\ &\bar{\alpha}_{i}^{G}G_{i}(\tilde{x}) \leq \bar{\alpha}_{i}^{G}G_{i}(y^{*}), \ i \in L,& \end{array} $$

  this implies that

  $$ \begin{array}{@{}rcl@{}} &&\sum\limits_{i \in M}\bar{\delta}_{i}g_{i}(\tilde{x})+\sum\limits_{i \in P}\bar{\mu}_{i}h_{i}(\tilde{x})-\sum\limits_{i \in L}\bar{\alpha}_{i}^{H}H_{i}(\tilde{x})+\sum\limits_{i \in L}\bar{\alpha}_{i}^{G}G_{i}(\tilde{x})\\ &\leq& \sum\limits_{i \in M}\bar{\delta}_{i}g_{i}(y^{*})+\sum\limits_{i \in P}\bar{\mu}_{i}h_{i}(y^{*})-\sum\limits_{i \in L}\bar{\alpha}_{i}^{H}H_{i}(y^{*})+\sum\limits_{i \in L}\bar{\alpha}_{i}^{G}G_{i}(y^{*}). \end{array} $$

  By the higher order strong quasiinvexity of $\sum \limits _{i \in M}\bar {\delta }_{i}g_{i}(.)+\sum \limits _{i \in P}\bar {\mu }_{i}h_{i}(.)-\sum \limits _{i \in L}\bar {\alpha }_{i}^{H}H_{i}(.)+\sum \limits _{i \in L}\bar {\alpha }_{i}^{G}G_{i}(.),$ with respect to the common kernel function ξ, one has

  $$ \begin{array}{@{}rcl@{}} \Bigg\langle \sum\limits_{i \in M}\bar{\delta}_{i}\nabla &g_{i}(y^{*})+\sum\limits_{i \in P}\bar{\mu}_{i} \nabla h_{i}(y^{*})-\sum\limits_{i \in L}\bar{\alpha}_{i}^{H} \nabla H_{i}(y^{*})+\sum\limits_{i \in L}\bar{\alpha}_{i}^{G} \nabla G_{i}(y^{*}), \xi (\tilde{x},y^{*}) \Bigg \rangle & \\& \!\!\!\!+c_{i} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma}+ c_{i} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} - {c_{i}^{H}} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma}+ {c_{i}^{G}} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} \leq 0.& \end{array} $$

  (3.24)

  Adding the inequalities (3.23) and (3.24), one has

  $$\langle \nabla L(y^{*}, \bar{\delta},\bar{ \mu}, \bar{\alpha}^{H}, \bar{\alpha}^{G}), \xi (\tilde{x}, y^{*}) \rangle +c \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} <0$$

  this is a contradiction to (3.21) and hence the result is proved.

• (2) Since $\tilde {x} \in X$ and $(y^{*}, \bar {\delta },\bar { \mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G},\bar {\rho }, \bar {\nu }) \in S_{MW}$, one has

  $$ \begin{array}{@{}rcl@{}} &\bar{\delta}_{i}g_{i}(\tilde{x}) \leq \bar{\delta}_{i}g_{i}(y^{*}), \ i \in M,&\\ &\bar{\mu}_{i}h_{i}(\tilde{x}) = \bar{\mu}_{i}h_{i}(y^{*}), \ i \in P, &\\ &-\bar{\alpha}_{i}^{H}H_{i}(\tilde{x}) \leq -\bar{\alpha}_{i}^{H}H_{i}(y^{*}), \ i \in L,&\\ &\bar{\alpha}_{i}^{G}G_{i}(\tilde{x}) \leq \bar{\alpha}_{i}^{G}G_{i}(y^{*}), \ i \in L.& \end{array} $$

  Using the above inequalities and (2.1), it follows that

  $$ \begin{array}{@{}rcl@{}} &&g_{i}(\tilde{x}) \leq g_{i}(y^{*}), \ i \in l_{g}^{+}(\tilde{x}),\\ &&h_{i}(\tilde{x}) = h_{i}(y^{*}), \ i \in l_{h}^{+}(\tilde{x}) \cup l_{h}^{-}(\tilde{x}),\\ &&-H_{i}(\tilde{x}) \leq -H_{i}(y^{*}), \ i \in l_{+0}^{+}(\tilde{x}) \cup l_{+-}^{+}(\tilde{x}) \cup l_{00}^{+}(\tilde{x}) \cup l_{0-}^{+}(\tilde{x}) \cup l_{0+}^{+}(\tilde{x}),\\ &&-H_{i}(\tilde{x}) \geq -H_{i}(y^{*}), \ i \in l_{0+}^{-}(\tilde{x}),\\ &&G_{i}(\tilde{x}) \leq G_{i}(y^{*}), \ i \in l_{+0}^{++}(\tilde{x}) \cup l_{+-}^{++}(\tilde{x}), \end{array} $$

  by the higher order strong quasiinvexity of $g_{i}(i \in l_{g}^{+}(\tilde {x})), h_{i}(i \in l_{h}^{+}(\tilde {x})),-h_{i}(i \in l_{h}^{-}(\tilde {x})), -H_{i}(i \in l_{+0}^{+}(\tilde {x})\cup l_{+-}^{+}(\tilde {x})\cup l_{00}^{+}(\tilde {x})\cup l_{0-}^{+}(\tilde {x})\cup l_{0+}^{+}(\tilde {x})),H_{i} (i \in l_{0+}^{-}(\tilde {x})), G_{i}(i \in l_{+0}^{++}(\tilde {x})\cup l_{+-}^{++}(\tilde {x})),$ with respect to the common kernel function ξ, it implies that

  $$ \begin{array}{@{}rcl@{}} &\langle \nabla g_{i}(y^{*}), \xi (\tilde{x},y^{*}) \rangle + c_{i} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} \leq 0, c_{i}>0, \bar{\delta}_{i}>0, \ i \in l_{g}^{+}(\tilde{x}), &\\ &\langle \nabla h_{i}(y^{*}), \xi (\tilde{x},y^{*}) \rangle +c_{i} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} \leq 0, c_{i}>0, \bar{\mu}_{i}>0, \ i \in l_{h}^{+}(\tilde{x}), &\\ &\langle \nabla h_{i}(y^{*}), \xi (\tilde{x},y^{*}) \rangle +c_{i} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} \geq 0, c_{i}>0, \bar{\mu}_{i}<0, \ i \in l_{h}^{-}(\tilde{x}), &\\ &-\langle \nabla H_{i}(y^{*}), \xi (\tilde{x},y^{*}) \rangle -{c_{i}^{H}} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} \leq 0, {c_{i}^{H}}>0, \bar{\alpha}_{i}^{H} \geq 0, \ i \in l_{+0}^{+}(\tilde{x}) \cup l_{+-}^{+}(\tilde{x}) \cup l_{00}^{+}(\tilde{x}) \cup &\\ & l_{0-}^{+}(\tilde{x}) \cup l_{0+}^{+}(\tilde{x}), &\\ &-\langle \nabla H_{i}(y^{*}), \xi (\tilde{x},y^{*}) \rangle -{c_{i}^{H}} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} \geq 0, {c_{i}^{H}}>0, \bar{\alpha}_{i}^{H} \leq 0, \ i \in l_{+0}^{-}(\tilde{x}), &\\ &\langle \nabla G_{i}(y^{*}), \xi (\tilde{x},y^{*}) \rangle +{c_{i}^{G}} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} \leq 0, {c_{i}^{G}}>0, \bar{\alpha}_{i}^{G} \geq 0, \ i \in l_{+0}^{++}(\tilde{x}) \cup l_{+-}^{++}(\tilde{x}).& \end{array} $$

  From the above inequalities and (2.1), it follows that

  $$ \begin{array}{@{}rcl@{}} &&\Bigg\langle \sum\limits_{i \in M}\delta_{i}\nabla g_{i} (y^{*})+\sum\limits_{i \in P}\mu_{i} \nabla h_{i}(y^{*})-\sum\limits_{i \in L}{\alpha_{i}^{H}} \nabla H_{i}(y^{*})+\sum\limits_{i \in L}{\alpha_{i}^{G}} \nabla G_{i}(y^{*}), \xi (\tilde{x},y^{*}) \Bigg\rangle\\ &&+c_{i} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma}+c_{i} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma}-{c_{i}^{H}} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma}+{c_{i}^{G}} \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} \leq 0. \end{array} $$

  Combining the above inequality and (3.21), one has

  $$\langle \nabla f(y^{*}), \xi(\tilde{x},y^{*}) \rangle +c \Vert \xi(\tilde{x},y^{*}) \Vert^{\sigma} \geq 0.$$

  By the higher order strong pseudoinvexity of f, with respect to the kernel function ξ, it follows that

  $$f (\tilde{x}) \geq f(y^{*})$$

  and hence the result is proved.

□

The following restricted converse duality theorem gives a sufficient condition for a feasible point of the MPVC to be a global minimum by utilizing the Mond–Weir dual.

Theorem 3.9

Let x^∗∈ X and $(y^{*}, \bar {\delta },\bar { \mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G},\bar {\rho }, \bar {\nu }) \in S_{MW}$ be feasible points for the MPVC and the VC-MWD, respectively, such that f(x^∗) = f(y^∗). If the hypothesis of Theorem 3.6 holds at y^∗∈ X ∪ prS_MW, then x^∗ is a global minimum of the MPVC.

Proof 9

Suppose to the contrary that x^∗∈ X is not a global minimum of the MPVC, then there exists $\tilde {x} \in X$ such that

$$f(\tilde{x}) \leq f (x^{*}).$$

From the assumptions in the theorem, it follows that

$$f(\tilde{x}) \leq f (y^{*}),$$

this is a contradiction to the Theorem 3.6 and hence the result is proved. □

The following strict converse duality theorem gives a sufficient condition about the uniqueness of a local minimum of the MPVC and a global maximum of the Wolfe dual model.

Theorem 3.10

Let x^∗∈ X be a local minimum for the MPVC such that the VC-ACQ holds at x^∗. Assume the conditions of Theorem 3.7 hold and $(y^{*}, \tilde {\delta }, \tilde {\mu }, \tilde {\alpha }^{H}, \tilde {\alpha }^{G},\tilde {\rho }, \tilde {\nu })$ be a global maximum of the VC-WD(x^∗). If one of the following conditions hold:

• (1) f(.) strictly higher order strongly pseudoinvex and $\sum \limits _{i \in M}\delta _{i}g_{i}(.)+\sum \limits _{i \in P}\mu _{i}h_{i}(.)-\sum \limits _{i \in L}{\alpha _{i}^{H}}H_{i}(.)+\sum \limits _{i \in L}{\alpha _{i}^{G}}G_{i}(.)$ is quasiinvex at y ∈ X ∪ prS_MW, with respect to the common kernel function ξ;

• (2) f(.) is strictly higher order strongly pseudoinvex and $g_{i}(i \in l_{g}^{+}(x^{*})), h_{i}(i \in l_{h}^{+}(x^{*})),-h_{i}(i \in l_{h}^{-}(x^{*})), -H_{i}(i \in l_{+0}^{+}(x^{*})\cup l_{+-}^{+}(x^{*})\cup l_{00}^{+}(x^{*})\cup l_{0-}^{+}(x^{*})\cup l_{0+}^{+}(x^{*})),H_{i} (i \in l_{0+}^{-}(x^{*})), G_{i}(i \in l_{+0}^{++} (x^{*})\cup l_{+-}^{++}(x^{*}))$ are strictly higher order strongly quasiinvex at y^∗∈ X ∪ prS_MW with respect to the common kernel function ξ;

Then x^∗ = y^∗.

Proof 10

• (1) Suppose that x^∗≠y^∗. By Theorem 3.7, there exist Lagrange multipliers $\bar {\delta } \in \mathbb {R}^{m}, \bar {\mu } \in \mathbb {R}^{p}, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu } \in \mathbb {R}^{l},$ such that $(y^{*},\bar {\delta }, \bar {\mu }, \bar {\alpha }^{H}, \bar {\alpha }^{G}, \bar {\rho }, \bar {\nu })$ be global maximum of the VC-MWD(x^∗). Hence,

  $$ \begin{array}{@{}rcl@{}} f(x^{*}) = f(y^{*}). \end{array} $$

  (3.25)

  Since x^∗∈ X and $(y^{*}, \tilde {\delta }, \tilde {\mu }, \tilde {\alpha }^{H}, \tilde {\alpha }^{G},\tilde {\rho }, \tilde {\nu }) \in S_{MW}$, it follows that

  $$ \begin{array}{@{}rcl@{}} &&g_{i}(x^{*})\leq 0, \ \tilde{\delta}_{i} \geq 0, \ i \in M,\\ &&h_{i}(x^{*})=0, \ \tilde{\mu}_{i} \in \mathbb{R}, \ i \in P,\\ &&-H_{i}(x^{*})<0, \ \tilde{\alpha}_{i}^{H} \geq 0, \ i \in l_{+}(x^{*}),\\ &&-H_{i}(x^{*})=0, \ \tilde{\alpha}_{i}^{H} \in \mathbb{R}, \ i \in l_{0}(x^{*}),\\ &&G_{i}(x^{*})>0,\ \tilde{\alpha}_{i}^{G} =0, \ i \in l_{0+}(x^{*}),\\ &&G_{i}(x^{*})=0, \ \tilde{\alpha}_{i}^{G} \geq 0, \ i \in l_{00}(x^{*}) \cup l_{+0}(x^{*}),\\ &&G_{i}(x^{*})<0, \ \tilde{\alpha}_{i}^{G} \geq 0, \ i \in l_{0-}(x^{*}) \cup l_{+-}(x^{*}). \end{array} $$

  By (3.21), it implies that

  $$ \begin{array}{@{}rcl@{}} &&\sum\limits_{i \in M}\tilde{\delta}_{i}g_{i}(x^{*})+\sum\limits_{i \in P}\tilde{\mu}_{i}h_{i}(x^{*})-\sum\limits_{i \in L}\tilde{\alpha}_{i}^{H}H_{i}(x^{*})+\sum\limits_{i \in L}\tilde{\alpha}_{i}^{G}G_{i}(x^{*})\\ &\leq& \sum\limits_{i \in M}\tilde{\delta}_{i}g_{i}(y^{*})+\sum\limits_{i \in P}\tilde{\mu}_{i}h_{i}(y^{*})-\sum\limits_{i \in L}\tilde{\alpha}_{i}^{H}H_{i}(y^{*})+\sum\limits_{i \in L} \tilde{\alpha}_{i}^{G}G_{i}(y^{*}). \end{array} $$

  Combining the higher order strong quasiinvexity of $\sum \limits _{i \in M}\tilde {\delta }_{i}g_{i}(.)+\sum \limits _{i \in P}\tilde {\mu }_{i}h_{i}(.)-\sum \limits _{i \in L}\tilde {\alpha }_{i}^{H}H_{i}(.)+\sum \limits _{i \in L}\tilde {\alpha }_{i}^{G}G_{i}(.)$ with respect to the kernel function ξ, one has

  $$ \begin{array}{@{}rcl@{}} \Bigg\langle \sum\limits_{i \in M}\tilde{\delta}_{i}&\nabla g_{i}(y^{*})+\sum\limits_{i \in P}\tilde{\mu}_{i}\nabla h_{i}(y^{*})-\sum\limits_{i \in L}\tilde{\alpha}_{i}^{H} \nabla H_{i}(y^{*})+\sum\limits_{i \in L} \tilde{\alpha}_{i}^{G} \nabla G_{i}(y^{*}) , \xi(x^{*}, y^{*}) \Bigg\rangle & \\& +c_{i} \Vert \xi(x^{*}, y^{*}) \Vert^{\sigma}+c_{i} \Vert \xi(x^{*}, y^{*}) \Vert^{\sigma}-{c_{i}^{H}} \Vert \xi(x^{*}, y^{*}) \Vert^{\sigma}+{c_{i}^{G}} \Vert \xi(x^{*}, y^{*}) \Vert^{\sigma} \leq 0.& \end{array} $$

  Using the above inequality and the first equation in (3.21), one has

  $$\langle \nabla f(y^{*}), \xi (x^{*}, y^{*}) \rangle +c \Vert \xi(x^{*}, y^{*}) \Vert^{\sigma} \geq 0.$$

  By the strictly higher order strong pseudoinvexity of f with respect to the kernel function ξ, we have

  $$f(x^{*}) > f(y^{*}).$$

  This is a contradiction to (3.25) and hence the result is proved.

• (2) Using x^∗∈ X and $(y^{*}, \tilde {\delta }, \tilde {\mu }, \tilde {\alpha }^{H}, \tilde {\alpha }^{G},\tilde {\rho }, \tilde {\nu }) \in S_{MW}(x^{*})$, it follows that

  $$ \begin{array}{@{}rcl@{}} &&g_{i}(x^{*}) \leq g_{i}(y^{*}), \ i \in l_{g}^{+}(x^{*}),\\ &&h_{i}(x^{*}) = h_{i}(y^{*}), \ i \in l_{h}^{+}(x^{*}) \cup l_{h}^{-}(x^{*}),\\ &&-H_{i}(x^{*}) \leq - H_{i}(y^{*}), \ i \in l_{+0}^{+}(x^{*}) \cup l_{+-}^{+}(x^{*}) \cup l_{00}^{+}(x^{*}) \cup l_{0-}^{+}(x^{*}) \cup l_{0+}^{+}(x^{*}),\\ &&-H_{i}(x^{*}) \geq \!-H_{i}(y^{*}), \ i \in l_{0+}^{-}(x^{*}),\\ &&G_{i}(x^{*}) \leq G_{i}(y^{*}), \ i \in l_{+0}^{++}(x^{*}) \cup l_{+-}^{++}(x^{*}). \end{array} $$

  By the higher order strong quasiinvexity of $g_{i}(i \in l_{g}^{+}(x^{*})), h_{i}(i \in l_{h}^{+}(x^{*})),-h_{i}(i \in l_{h}^{-}(x^{*})), -H_{i}(i \in l_{+0}^{+}(x^{*})\cup l_{+-}^{+}(x^{*})\cup l_{00}^{+}(x^{*})\cup l_{0-}^{+}(x^{*})\cup l_{0+}^{+}(x^{*})),H_{i} (i \in l_{0+}^{-}(x^{*})), G_{i}(i \in l_{+0}^{++}(x^{*})\cup l_{+-}^{++}(x^{*}))$ with respect to the common kernel function ξ, it implies that

  $$ \begin{array}{@{}rcl@{}} &\langle \nabla g_{i}(y^{*}), \xi (x^{*},y^{*}) \rangle +c_{i} \Vert \xi(x^{*}, y^{*}) \Vert^{\sigma} \leq 0, c_{i}>0, \tilde{\delta}_{i}>0, \ i \in l_{g}^{+}(x^{*}), &\\ &\langle \nabla h_{i}(y^{*}), \xi (x^{*},y^{*}) \rangle +c_{i} \Vert \xi(x^{*}, y^{*}) \Vert^{\sigma} \leq 0, c_{i}>0, \tilde{\mu}_{i}>0, \ i \in l_{h}^{+}(x^{*}), &\\ &\langle \nabla h_{i}(y^{*}), \xi (x^{*},y^{*}) \rangle +c_{i} \Vert \xi(x^{*}, y^{*}) \Vert^{\sigma} \geq 0, c_{i}>0, \tilde{\mu}_{i}<0, \ i \in l_{h}^{-}(x^{*}), &\\ &-\langle \nabla H_{i}(y^{*}), \xi (x^{*},y^{*}) \rangle -{c_{i}^{H}} \Vert \xi(x^{*}, y^{*}) \Vert^{\sigma} \leq 0, {c_{i}^{H}}>0, \tilde{\alpha}_{i}^{H} \geq 0, \ i \in l_{+0}^{+}(x^{*}) \cup l_{+-}^{+}(x^{*}) \cup l_{00}^{+}(x^{*}) &\\& \cup l_{0-}^{+}(x^{*}) \cup l_{0+}^{+}(x^{*}), &\\ &-\langle \nabla H_{i}(y^{*}), \xi (x^{*},y^{*}) \rangle -{c_{i}^{H}} \Vert \xi(x^{*}, y^{*}) \Vert^{\sigma} \geq 0, {c_{i}^{H}}>0, \tilde{\alpha}_{i}^{H}, \leq 0, \ i \in l_{+0}^{+}(x^{*}), &\\ &\langle \nabla G_{i}(y^{*}), \xi (x^{*},y^{*}) \rangle +{c_{i}^{G}} \Vert \xi(x^{*}, y^{*}) \Vert^{\sigma} \leq 0, {c_{i}^{G}}>0, \tilde{\alpha}_{i}^{G} \geq 0, \ i \in l_{+0}^{++}(x^{*}) \cup l_{+-}^{++}(x^{*}).& \end{array} $$

  From the above inequalities and (2.1), it follows that

  $$ \begin{array}{@{}rcl@{}} &&\Bigg\langle \sum\limits_{i \in M}\tilde{\delta}_{i}\nabla g_{i}(y^{*})+\sum\limits_{i \in P}\tilde{\mu}_{i} \nabla h_{i}(y^{*})-\sum\limits_{i \in L}\tilde{\alpha}_{i}^{H} \nabla H_{i}(y^{*})+\sum\limits_{i \in L}\tilde{\alpha}_{i}^{G} \nabla G_{i}(y^{*}), \xi (x^{*},y^{*}) \Bigg\rangle\\ &&+c_{i} \Vert \xi(x^{*}, y^{*}) \Vert^{\sigma}+c_{i} \Vert \xi(x^{*}, y^{*}) \Vert^{\sigma}-{c_{i}^{H}} \Vert \xi(x^{*}, y^{*}) \Vert^{\sigma}+{c_{i}^{G}} \Vert \xi(x^{*}, y^{*}) \Vert^{\sigma} \leq 0. \end{array} $$

  Combining the above inequality and (3.21), one has

  $$\langle \nabla f(y^{*}), \xi(x^{*},y^{*}) \rangle +c \Vert \xi(x^{*}, y^{*}) \Vert^{\sigma}\geq 0.$$

  By the higher order strong pseudoinvexity of f, with respect to the kernel function ξ it implies that

  $$f (x^{*}) \geq f(y^{*}).$$

  This is a contradiction to (3.25) and hence the result is proved.

□

Example 3.2

Consider the MPVC of Example 3.1. For any feasible x ∈ X,the VC-MWD(x) to the MPVC is given by

$$ \begin{array}{@{}rcl@{}} \max &&f(y) = {y_{1}^{2}}-{y_{2}^{2}}\\ && s.t. \nabla L(y, {\alpha_{1}^{H}}, {\alpha_{1}^{G}})=\left( 2y_{1}- {\alpha_{1}^{H}} + {\alpha_{1}^{G}}, -2y_{2}+2y_{2}{\alpha_{1}^{H}}\!+2y_{2}{\alpha_{1}^{G}} \right)=(0,0),\\ &&{\alpha_{1}^{G}}G_{1}(y)={\alpha_{1}^{G}}(y_{1}+{y_{2}^{2}})\geq 0,\\ &&{\alpha_{1}^{G}} = \nu_{1}(x_{1}-{x_{2}^{2}}), \nu_{1} \geq 0,\\ &&-{\alpha_{1}^{H}}H_{1}(y)=-{\alpha_{1}^{H}}(y_{1}-{y_{2}^{2}}) \geq 0,\\ &&{\alpha_{1}^{H}}= \rho_{1}-\nu_{1}(x_{1}+{x_{2}^{2}}), \rho_{1} \geq 0. \end{array} $$

(3.26)

• (1) Let x^∗ = (0,0) and $(y^{*}, {\alpha _{1}^{G}}, {\alpha _{1}^{H}}, \rho _{1}, \nu _{1})=(0,0,0,0,0) \in S_{W}(x^{*}),$ that is y^∗ := (0,0) ∈ prS_MW. We have

  $$f(x^{*})=0=f(y^{*}).$$

  It can be verified that the hypothesis of Theorem 3.8 holds. Taking into account (3.19), x^∗ is a global minimum of (3.19). So, Theorem 3.8 is verified.

• (2) We can get $y_{1}=\frac {{\alpha _{1}^{H}}-{\alpha _{1}^{G}}}{2}$ and for ${\alpha _{1}^{H}} + {\alpha _{1}^{G}} \neq 1$ we have y₂ = 0. From (3.26), one also has ${\alpha _{1}^{G}}=0$ and ${\alpha _{1}^{H}}\geq 0$, i.e.,

  $$ L(y_{1},y_{2}, {\alpha_{1}^{H}}, {\alpha_{1}^{G}})=f(y)-{\alpha_{1}^{H}}H_{1}(y)+{\alpha_{1}^{G}}G_{1}(y)={y_{1}^{2}}-{y_{2}^{2}}-{\alpha_{1}^{H}}H_{1}(y)+{\alpha_{1}^{G}}G_{1}(y)=-\frac{1}{2}({\alpha_{1}^{H}})^{2}. $$

  We obtain, $L(y_{1},y_{2}, {\alpha _{1}^{H}}, {\alpha _{1}^{G}}) \leq 0.$ This implies that

  $$f(y)\leq {\alpha_{1}^{H}}H_{1}(y)-{\alpha_{1}^{G}}G_{1}(y).$$

  Combining (3.26), we can get f(y) ≤ 0. So we obtain f(x) ≥ f(y). Theorem (3.6) is verified.

• (3) Since x^∗ = (0,0) is the unique solution of MPVC and ∇H₁ = (1,0)^T,∇G₁ = (1,0)^T. It is easy to see that (3.19) satisfies VC-ACQ. By Theorem 2.1, there exist Lagrange multipliers ${\alpha _{1}^{H}}, {\alpha _{1}^{G}}, \rho _{1}, \nu _{1} \in \mathbb {R}$ such that $(0, {\alpha _{1}^{H}}, {\alpha _{1}^{G}}, \rho _{1}, \nu _{1})$ is a feasible point of the VC-WD(0). Taking into account f(y) ≤ 0, we get $(0, {\alpha _{1}^{H}}, {\alpha _{1}^{G}}, \rho _{1}, \nu _{1})$ is a global maximum of the VC-MWD(0) and Theorem 3.7 is verified.

4 Conclusions

In this paper, we have established the weak, strong, converse and restricted converse duality results under the assumptions of higher order strong invexity, strict invexity, pseudoinvexity, strict pseudoinvexity and quasiinvexity. Examples are also given in order to verify the results. Further, some other dual models for the primal MPVC, like the mixed type dual may be investigated by using the univexity and generalized univexity assumptions to obtain the duality results. However, some interesting topics for further research remain. Also, it would be interesting to prove similar optimality and duality results for multiobjective optimization problems. We shall investigate these questions in subsequent papers.