A Duality Framework for Mathematical Programs with Tangential Subdifferentials

Abstract

The aim of this article is to study duality results for nonsmooth mathematical programs with equilibrium constraints in terms of tangential subdifferentials. We study the Wolfe-type dual problem under the convexity assumptions and a Mond–Weir-type dual problem is also formulated under convexity and generalized convexity assumptions for MPEC by using tangential subdifferentials. We establish weak duality and the two dual programs by assuming tangentially convex functions and also obtain strong duality theorems by assuming generalized standard Abadie constraint qualification.

1. Introduction

Pshenichnyi [1] pioneered the idea of a specific category of functions known as tangentially convex. Later, Lemar’echal [2] conducted extensive research on this topic and is credited with popularizing the term “tangentially convex.” This class of functions encompasses various types, including Clark regular functions, Gateaux differentiable functions, and Michel–Penot regular subdifferentials [2,3]. An intriguing subset of nonlinear programming problems involves mathematical programming problems with equilibrium constraints (MPECs). These problems represent a unique and noteworthy area within the broader field of optimization. MPEC refers to a class of optimization problems in which part of the feasible set is determined implicitly by equilibrium conditions. These constraints can take the form of complementarity systems, variational inequalities, or other equilibrium formulations. Many authors gave several different approaches to solve this problem for the smooth case [4,5,6] and the nonsmooth case [7,8,9]. In 2015, Guo et al. [10] gave effective numerical methods for solving MPEC. MPEC arises frequently in various fields such as chemical process engineering, economic equilibrium, multilevel games, etc. [11]. MPEC problems pose significant challenges in terms of solution methods. Geometrically, the feasible region of such problems is typically non-convex and often lacks connectivity. Additionally, the constraints in MPEC typically violate the standard constraint qualifications—such as the Linear Independence Constraint Qualification (LICQ) and the Mangasarian–Fromovitz Constraint Qualification (MFCQ)—at all feasible points. To address these complexities, numerous researchers have proposed various stationarity concepts and constraint qualifications tailored to MPEC [12,13,14]. Recently, Mishra and Singh advanced the field by introducing several constraint qualifications and stationarity conditions utilizing the framework of tangential subdifferentials [15].

In nonlinear programming problems, the concept of duality is very important. Many authors studied dual problems corresponding to primal problems for solving different optimization problems in various fields (see [16,17,18]). For application of duality (see [19,20,21]), in 1961, Wolfe [22] introduced Wolfe duality, and in 1981, Mond and Weir [23] introduced Mond–Weir duality for different scalar functions. Later, many authors extended these duality results for nondifferentiable functions by using the tool of generalized convexity (see [24,25,26,27,28]). Tung [29] established KKT optimality conditions and duality for multiobjective semi-infinite programming by using tangential subdifferentials.

Notably, duality results for MPEC in the sense of tangential subdifferentials have not been studied yet. Motivated by the aforementioned contributions, we examine dual programs of the Wolfe-type and Mond–Weir-type for mathematical programming problems with equilibrium constraints (MPECs). In this context, we explore a range of duality theorems, employing the concept of tangential subdifferentials.

The organization of this paper is as follows: Section 2 provides some basic definitions and theorems that will be needed in the sequel of the paper. Section 3 presents Wolfe and Mond–Weir dual problems and establish weak and strong duality theorems for mathematical programs with equilibrium constraints and the two dual models using generalized convexity in the sense of tangential subdifferentials.

2. Preliminaries

In this section, we discuss some basic definitions and concepts that will be used in this sequel.

Let $S\subset {\mathbb{R}}^n$ be a non-empty set. We denote by $coS$ the convex hull of S and by cone S the convex cone generated by S. The notation $-S$ represents the negative polar cone, defined as

$$\Lambda (\aleph ,S)=\{\kappa \in {\mathbb{R}}^n:\exists t_n\downarrow 0,\exists {\kappa }_n\to \kappa suchthat\aleph +t_n{\kappa }_n\in S\}.$$

The closure of a set S, written as $clS$, is the smallest closed set containing S. For a point $\aleph \in clS$, the contingent cone (or Bouligand tangent cone) to S at ℵ is given by

$$\Lambda (\aleph ,S)=\{\kappa \in {\mathbb{R}}^n:\exists t_n\downarrow 0,\exists {\kappa }_n\to \kappa suchthat\aleph +t_n{\kappa }_n\in S\}.$$

Our objective is to study the following mathematical programming problem with equilibrium constraints (MPEC):

$$\begin{array}{cc}\hfill \left(P\right):\min \mathcal{P}(\aleph )& \\ \hfill \mathrm{subject}\mathrm{to}\mathcal{Q}(\aleph )& \le 0,\mathcal{R}(\aleph )=0,\hfill \\ \hfill \mathcal{S}(\aleph )& \ge 0,\mathcal{T}(\aleph )\ge 0,⟨\mathcal{S}(\aleph ),\mathcal{T}(\aleph )⟩=0,\hfill \end{array}$$

where $\mathcal{P}:{\mathbb{R}}^n\to \mathbb{R}$, $\mathcal{Q}:{\mathbb{R}}^n\to {\mathbb{R}}^k$, $\mathcal{R}:{\mathbb{R}}^n\to {\mathbb{R}}^p$, $\mathcal{S}:{\mathbb{R}}^n\to {\mathbb{R}}^l$ and $\mathcal{T}:{\mathbb{R}}^n\to {\mathbb{R}}^l$ are tangentially convex functions and n = 1,2…N, p = 1,2…N, k = 1,2…N, l = 1,2…N where N is a finite number and n, p, k, l $\in \mathbb{N}$. The feasibility set of $\left(P\right)$ is denoted by X, that is,

$$X=\{\aleph \in {\mathbb{R}}^n:\mathcal{Q}(\aleph )\le 0,\mathcal{R}(\aleph )=0,\mathcal{S}(\aleph )\ge 0,\mathcal{T}(\aleph )\ge 0,⟨\mathcal{S}(\aleph ),\mathcal{T}(\aleph )⟩=0\}.$$

Index sets for a feasible vector $\overline{\aleph }\in X$ for the problem $\left(P\right)$.

$$\begin{array}{cc}\hfill I_{\mathcal{Q}}& :=I_{\mathcal{Q}}\left(\overline{\aleph }\right)=\{r=1,2,\dots ,k:{\mathcal{Q}}_r\left(\overline{\aleph }\right)=0\}\hfill \\ \hfill \Theta & :=\Theta \left(\overline{\aleph }\right)=\{r=1,2\dots ,l:{\mathcal{S}}_r\left(\overline{\aleph }\right)=0,{\mathcal{T}}_r\left(\overline{\aleph }\right)>0\},\hfill \\ \hfill \Omega & :=\mathrm{\Omega }\left(\overline{\aleph }\right)=\{r=1,2,\dots ,l:{\mathcal{S}}_r\left(\overline{\aleph }\right)=0,{\mathcal{T}}_r\left(\overline{\aleph }\right)=0\},\hfill \\ \hfill \mathrm{{\rm Y}}& :=\mathrm{{\rm Y}}\left(\overline{\aleph }\right)=\{r=1,2,\dots ,l:{\mathcal{S}}_r\left(\overline{\aleph }\right)>0,{\mathcal{T}}_r\left(\overline{\aleph }\right)=0\},\hfill \end{array}$$

where $\Omega$ is called the degenerate set.

Definition 1.

A function $\mathcal{R}:{\mathbb{R}}^n\to \mathbb{R}$ is called directionally differentiable at a point $\overline{\aleph }\in {\mathbb{R}}^n$ in the direction $d\in {\mathbb{R}}^n$ if the limit

$${\mathcal{R}}^{\prime }(\overline{\aleph };d)=\underset{\alpha \to 0}{lim}{\displaystyle \frac{\mathcal{R}(\overline{\aleph }+\alpha d)-\mathcal{R}\left(\overline{\aleph }\right)}{\alpha }}$$

exists and is finite. Furthermore, $\mathcal{R}$ is said to be directionally differentiable (or semi-differentiable) at $\overline{\aleph }$ if the directional derivative ${\mathcal{R}}^{\prime }(\overline{\aleph };d)$ exists and is finite for all directions $d\in {\mathbb{R}}^n$.

Definition 2

([2,3,30]). A function $\Phi :{\mathbb{R}}^n\to \mathbb{R}$ is said to be tangentially convex at a point $\overline{\aleph }\in {\mathbb{R}}^n$ if the directional derivative ${\Phi }^{\prime }(\overline{\aleph },d)$ exists and is finite for every direction $d\in {\mathbb{R}}^n$. Moreover, the mapping

$$d\to {\Phi }^{\prime }(\overline{\aleph },d),d\in {\mathbb{R}}^n$$

is required to be a convex function of d.

Definition 3

([1,3]). Let $\Phi :{\mathbb{R}}^n\to \mathbb{R}$ be tangentially convex at a point $\overline{\aleph }\in {\mathbb{R}}^n$. The tangential subdifferential of Φ at $\overline{\aleph }$, denoted by ${\partial }^T\Phi \left(\overline{\aleph }\right)$, is a non-empty, compact, and convex subset of ${\mathbb{R}}^n$ such that the directional derivative satisfies

$${\Phi }^{\prime }(\overline{\aleph };d)=\underset{{\aleph }^{*}\in {\partial }^T\Phi \left(\overline{\aleph }\right)}{max}⟨{\aleph }^{*},d⟩,\forall d\in {\mathbb{R}}^n.$$

Equivalently, it can be represented as

$${\partial }^T\Phi \left(\overline{\aleph }\right)=\left\{{\aleph }^{*}\in {\mathbb{R}}^n\mid ⟨{\aleph }^{*},d⟩\le {\Phi }^{\prime }(\overline{\aleph };d),\forall d\in {\mathbb{R}}^n\right\}.$$

Now, we prove ${\partial }^T\Phi \left(\overline{\aleph }\right)$ is convex and compact.

Assuming $\Phi$ is tangentially convex (so directional derivative ${\Phi }^{\prime }(\overline{\aleph };.)$ is sublinear) and that $\overline{\aleph }$ lies in a region where the domain is nice (interior of domain), one can show the following:

• Convexity: Because ${\Phi }^{\prime }(\overline{\aleph };.)$ is sublinear, the set of all linear functionals that lie below ${\Phi }^{\prime }(\overline{\aleph };.)$$(i.e.,\{{\overline{\aleph }}^{*}:⟨{\aleph }^{*},d⟩\le {\Phi }^{\prime }(\overline{\aleph };d)\forall d\})$ is a convex set. This is because the intersection of inequality-halfspaces of this form (one for each d) is convex.
• Closedness: The set is defined by an infinite family of linear inequalities ${\Phi }^{\prime }(\overline{\aleph };.)$$(i.e.,\{⟨{\aleph }^{*},d⟩\le {\Phi }^{\prime }(\overline{\aleph };d)\forall d\})$ Each such constraint defines a closed halfspace in ${\mathbb{R}}^{⋉}$. Intersection of closed halfspaces is closed.
• Boundedness (hence compactness in finite dimension): One typically needs a bound on growth of ${\Phi }^{\prime }(\overline{\aleph };.d)$ relative to d. Because ${\Phi }^{\prime }(\overline{\aleph };d)$ is positively homogeneous, sublinear, there is often a Lipschitz bound or some estimate near d vectors that ensures the supporting function is finite, restricting how large $x^{*}$ can be. Once the set is closed and bounded, in finite dimensions it is compact.

The following concept of various stationary points was introduced by Mishra and Singh [15] in terms of tangential subdifferentials.

Definition 4

([15]). GW (Generalised Weakly)-stationary point A feasible point $\overline{\aleph }$ of problem $\left(P\right)$ is called $GW$ a stationary point if there are vectors $\lambda =({\lambda }^{\mathcal{Q}},{\lambda }^{\mathcal{R}},{\lambda }^{\mathcal{S}},{\lambda }^{\mathcal{T}})\in {\mathbb{R}}^{k+p+2l}$ and $\mu =({\mu }^{\mathcal{R}},{\mu }^{\mathcal{S}},{\mu }^{\mathcal{T}})\in {\mathbb{R}}^{p+2l}$ such that

$$\begin{array}{cc}\hfill & 0\in co{\partial }^T{\mathcal{P}}_r\left(\overline{\aleph }\right)+\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}co{\partial }^T{\mathcal{Q}}_r\left(\overline{\aleph }\right)+\sum _{j=1}^p\left[{\lambda }_j^{\mathcal{R}}co{\partial }^T{\mathcal{R}}_j\left(\overline{\aleph }\right)+{\mu }_j^{\mathcal{R}}co{\partial }^T(-{\mathcal{R}}_j)\left(\overline{\aleph }\right)\right]\hfill \\ & +\sum _{r=1}^l{\lambda }_r^{\mathcal{S}}co{\partial }^T(-{\mathcal{S}}_r)\left(\overline{\aleph }\right)+{\lambda }_r^{\mathcal{T}}co{\partial }^T(-{\mathcal{T}}_r)\left(\overline{\aleph }\right)\hfill \\ & +\sum _{r=1}^l{\mu }_r^{\mathcal{S}}co{\partial }^T\left({\mathcal{S}}_r\right)\left(\overline{\aleph }\right)+{\mu }_r^{\mathcal{T}}co{\partial }^T\left({\mathcal{T}}_r\right)\left(\overline{\aleph }\right)\hfill \end{array}$$

(1)

$$\begin{array}{cc}& {\lambda }_{I_{\mathcal{Q}}}^{\mathcal{Q}}\ge 0,{\lambda }_j^{\mathcal{R}},{\mu }_j^{\mathcal{R}}\ge 0,j=1,\dots ,p,{\lambda }_r^{\mathcal{S}},{\lambda }_r^{\mathcal{T}},{\mu }_r^{\mathcal{S}},{\mu }_r^{\mathcal{T}}\ge 0,r=1,\dots ,l,\hfill \end{array}$$

(2)

$$\begin{array}{c}\hfill {\lambda }_{{\rm Y}}^{\mathcal{S}}={\lambda }_{\Theta }^{\mathcal{T}}={\mu }_{{\rm Y}}^{\mathcal{S}}={\mu }_{\Theta }^{\mathcal{T}}=0.\end{array}$$

(3)

Definition 5

([15]). GA (Generalised Alternatively)-stationary point A feasible point $\overline{\aleph }$ of problem $\left(P\right)$ is said to $GA$ be a stationary point if there are vectors $\lambda =({\lambda }^{\mathcal{Q}},{\lambda }^{\mathcal{R}},{\lambda }^{\mathcal{S}},{\lambda }^{\mathcal{T}})\in {\mathbb{R}}^{k+p+2l}$ and $\mu =({\mu }^{\mathcal{R}},{\mu }^{\mathcal{S}},{\mu }^{\mathcal{T}})\in {\mathbb{R}}^{p+2l}$ such that it satisfies the above conditions (1)–(3) together with the following condition:

$${\mu }_r^{\mathcal{S}}=0or{\mu }_r^{\mathcal{T}}=0,\forall r\in \Omega .$$

Definition 6

([15]). GS (Generalised Strong)-stationary point A feasible point $\overline{\aleph }$ of problem $\left(P\right)$ is said to $GS$-stationary point if, there are vectors $\lambda =({\lambda }^{\mathcal{Q}},{\lambda }^{\mathcal{R}},{\lambda }^{\mathcal{S}},{\lambda }^{\mathcal{T}})\in {\mathbb{R}}^{k+p+2l}$ and $\mu =({\mu }^{\mathcal{R}},{\mu }^{\mathcal{S}},{\mu }^{\mathcal{T}})\in {\mathbb{R}}^{p+2l}$ such that it satisfies the above conditions (1)–(3) together with following condition:

$${\mu }_r^{\mathcal{S}}=0,{\mu }_r^{\mathcal{T}}=0,\forall r\in \Omega .$$

Definition 7

([15]). Let $\mathcal{P}:{\mathbb{R}}^n\to \mathbb{R}\cup \{+\infty \}$ be an extended real-valued function and $\mathcal{P}$ is tangentially convex function at $\overline{\aleph }$. Then, $\mathcal{P}$ is said to be

(i) 

${\partial }^T$-convex at $\overline{\aleph }$, if and only if $\forall \aleph \in {\mathbb{R}}^n$, $\mathcal{P}(\aleph )\ge \mathcal{P}\left(\overline{\aleph }\right)+⟨\xi ,\aleph -\overline{\aleph }⟩,\forall \xi \in {\partial }^T\mathcal{P}\left(\overline{\aleph }\right)$;

(ii) 

${\partial }^T$-pseudoconvex at $\overline{\aleph }$, if and only if $\forall \aleph \in {\mathbb{R}}^n$, $\mathcal{P}(\aleph )<\mathcal{P}\left(\overline{\aleph }\right)\Rightarrow ⟨\xi ,\aleph -\overline{\aleph }⟩<0,\forall \xi \in {\partial }^T\mathcal{P}\left(\overline{\aleph }\right)$;

(iii) 

${\partial }^T$-quasiconvex at $\overline{\aleph }$ if and only if $\forall \aleph \in {\mathbb{R}}^n$, $\mathcal{P}(\aleph )\le \mathcal{P}\left(\overline{\aleph }\right)\Rightarrow ⟨\xi ,\aleph -\overline{\aleph }⟩\le 0,\forall \xi \in {\partial }^T\mathcal{P}\left(\overline{\aleph }\right)$.

3. Duality

In this section, we formulate a dual problem of the Wolfe type for the primal problem $\left(P\right)$, leveraging the concept of ${\partial }^T$-convexity. Additionally, we analyze a Mond–Weir-type dual problem under the assumptions of ${\partial }^T$-convexity and generalized ${\partial }^T$-convexity for the primal problem $\left(P\right)$.

The required index sets will be used:

$$\begin{array}{cc}\hfill & {\Omega }_{\mu }^{\mathcal{S}}=\{r\in \Omega :{\mu }_r^{\mathcal{T}}=0,{\mu }_r^{\mathcal{S}}>0\},{\Omega }_{\mu }^{\mathcal{T}}=\{r\in \Omega :{\mu }_r^{\mathcal{S}}=0,{\mu }_r^{\mathcal{T}}>0\},\hfill \\ & {\Theta }_{\mu }^{+}=\{r\in \Theta :{\mu }_r^{\mathcal{S}}>0\},{\mathrm{{\rm Y}}}_{\mu }^{+}=\{r\in \mathrm{{\rm Y}}:{\mu }_r^{\mathcal{T}}>0\}.\hfill \end{array}$$

$WD\left(\overline{\aleph }\right)$

$$\begin{array}{c}\hfill \underset{u,\lambda }{max}\left\{\mathcal{P}\left(u\right)+\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\mathcal{Q}}_r\left(u\right)+\sum _{j=1}^p{\theta }_j^{\mathcal{R}}{\mathcal{R}}_j\left(u\right)-\sum _{r=1}^l\left[{\lambda }_r^{\mathcal{S}}{\mathcal{S}}_r\left(u\right)+{\lambda }_r^{\mathcal{T}}{\mathcal{T}}_r\left(u\right)\right]\right\}\end{array}$$

subject to

$$\begin{array}{cc}\hfill & 0\in co{\partial }^T{\mathcal{P}}_r\left(u\right)+\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}co{\partial }^T{\mathcal{Q}}_r\left(u\right)+\sum _{j=1}^p\left[{\lambda }_j^{\mathcal{R}}co{\partial }^T{\mathcal{R}}_j\left(u\right)+{\mu }_j^{\mathcal{R}}co{\partial }^T(-{\mathcal{R}}_j)\left(u\right)\right]\hfill \\ & +\sum _{r=1}^l\left[{\lambda }_r^{\mathcal{S}}co{\partial }^T(-{\mathcal{S}}_r)\left(u\right)+{\lambda }_r^{\mathcal{S}}co{\partial }^T(-{\mathcal{T}}_r)\left(u\right)\right],\hfill \\ & {\lambda }_{I_{\mathcal{Q}}}^{\mathcal{Q}}\ge 0,{\lambda }_j^{\mathcal{R}},{\mu }_j^{\mathcal{R}}\ge 0,j=1,\dots ,p,{\lambda }_r^{\mathcal{S}},{\lambda }_r^{\mathcal{T}},{\mu }_r^{\mathcal{S}},{\mu }_r^{\mathcal{T}}\ge 0,r=1,\dots ,l,\hfill \\ & {\lambda }_{\mathrm{{\rm Y}}}^{\mathcal{S}}={\lambda }_{\Theta }^{\mathcal{T}}={\mu }_{{\rm Y}}^{\mathcal{S}}={\mu }_{\Theta }^{\mathcal{T}}=0,\forall r\in \Omega ,{\mu }_r^{\mathcal{S}}=0,{\mu }_r^{\mathcal{T}}=0,\hfill \end{array}$$

(4)

where

• ${\theta }_j^{\mathcal{R}}={\lambda }_j^{\mathcal{R}}-{\mu }_j^{\mathcal{R}},\lambda =({\lambda }^{\mathcal{Q}},{\lambda }^{\mathcal{R}},{\lambda }^{\mathcal{S}},{\lambda }^{\mathcal{T}})\in {\mathbb{R}}^{k+p+2l}$ and $\mu =({\mu }^{\mathcal{R}},{\mu }^{\mathcal{S}},{\mu }^{\mathcal{T}})\in {\mathbb{R}}^{p+2l}.$

Theorem 1

(Weak Duality). Suppose the primal problem $\left(P\right)$ and its dual $WD\left(\overline{\aleph }\right)$ and we take $\overline{\aleph }$ and $(u,\lambda )$ to be feasible for the primal and its dual, respectively.The index sets $I_{\mathcal{Q}},\Theta ,\Omega ,{\rm Y}$ are defined accordingly. We consider $\mathcal{P}$, ${\mathcal{Q}}_r(r\in I_{\mathcal{Q}}),\underline{+}{\mathcal{R}}_j(j=1,\dots ,p),-{\mathcal{S}}_r(r\in \Theta \cup \Omega ),-{\mathcal{T}}_r(r\in {\rm Y}\cup \Omega )$ as tangentially convex functions and are ${\partial }^T\text{-}convex$ functions at u. If ${\Omega }_{\mu }^{\mathcal{S}}\cup {\Omega }_{\mu }^{\mathcal{T}}\cup {\Theta }_{\mu }^{+}\cup {{\rm Y}}_{\mu }^{+}=\varnothing$, then for any ℵ feasible for the problem $\left(P\right)$, we have

$$\begin{array}{c}\hfill \mathcal{P}(\aleph )\ge \mathcal{P}\left(u\right)+\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\mathcal{Q}}_r\left(u\right)+\sum _{j=1}^p{\theta }_j^{\mathcal{R}}{\mathcal{R}}_j\left(u\right)-\sum _{r=1}^l\left[{\lambda }_r^{\mathcal{S}}{\mathcal{S}}_r\left(u\right)+{\lambda }_r^{\mathcal{T}}{\mathcal{T}}_r\left(u\right)\right].\end{array}$$

Proof. 

Suppose ℵ be any feasible point for the problem $\left(P\right)$. Then, we have

$${\mathcal{Q}}_r(\aleph )\le 0,\forall r\in I_{\mathcal{Q}}$$

and

$${\mathcal{R}}_j(\aleph )=0,j=1,2,\dots ,p.$$

Since $\mathcal{P}$ is convex at u, then,

$$\begin{array}{c}\hfill \mathcal{P}(\aleph )-\mathcal{P}\left(u\right)\ge ⟨\xi ,\aleph -u⟩,\forall \xi \in {\partial }^T\mathcal{P}\left(u\right).\end{array}$$

(5)

Similarly, we have

$$\begin{array}{c}\hfill {\mathcal{Q}}_r(\aleph )-{\mathcal{Q}}_r\left(u\right)\ge ⟨{\xi }_r^{\mathcal{Q}},\aleph -u⟩,\forall {\xi }_r^{\mathcal{Q}}\in {\partial }^T{\mathcal{Q}}_r\left(u\right),\forall r\in I_{\mathcal{Q}},\end{array}$$

(6)

$$\begin{array}{c}\hfill {\mathcal{R}}_j(\aleph )-{\mathcal{R}}_j\left(u\right)\ge ⟨{\eta }_j,\aleph -u⟩,\forall {\eta }_j\in {\partial }^T{\mathcal{R}}_j\left(u\right),\forall j=\{1,\dots ,p\},\end{array}$$

(7)

$$\begin{array}{c}\hfill -{\mathcal{R}}_j(\aleph )+{\mathcal{R}}_j\left(u\right)\ge ⟨{\nu }_j,\aleph -u⟩,\forall {\nu }_j\in {\partial }^T(-{\mathcal{R}}_j)\left(u\right),\forall j=\{1,\dots ,p\},\end{array}$$

(8)

$$\begin{array}{c}\hfill -{\mathcal{S}}_r(\aleph )+{\mathcal{S}}_r\left(u\right)\ge ⟨{\xi }_r^{\mathcal{S}},\aleph -u⟩,\forall {\xi }_r^{\mathcal{S}}\in {\partial }^T(-{\mathcal{S}}_r)\left(u\right),\forall r\in \Theta \cup \Omega ,\end{array}$$

(9)

$$\begin{array}{c}\hfill -{\mathcal{T}}_r(\aleph )+{\mathcal{T}}_r\left(u\right)\ge ⟨{\xi }_r^{\mathcal{T}},\aleph -u⟩,\forall {\xi }_r^{\mathcal{T}}\in {\partial }^T(-{\mathcal{T}}_r)\left(u\right),\forall r\in \Omega \cup {\rm Y}.\end{array}$$

(10)

If ${\Omega }_{\mu }^{\mathcal{S}}\cup {\Omega }_{\mu }^{\mathcal{T}}\cup {\Theta }_{\mu }^{+}\cup {{\rm Y}}_{\mu }^{+}=\varnothing ,$ then multiplying (6)–(10) by ${\lambda }_r^{\mathcal{Q}}\ge 0(r\in I_{\mathcal{Q}}),{\lambda }_j^{\mathcal{R}}>0(j=1,\dots ,p),{\mu }_j^{\mathcal{R}}>0(j=1,\dots ,p),{\lambda }_r^{\mathcal{S}}>0(r\in \Theta \cup \Omega ),{\lambda }_r^{\mathcal{T}}>0(r\in \mathrm{{\rm Y}}\cup \Omega )$, respectively, and adding (5)–(10), we get

$$\begin{array}{cc}\hfill & \mathcal{P}(\aleph )-\mathcal{P}\left(u\right)+\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\mathcal{Q}}_r(\aleph )-\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\mathcal{Q}}_r\left(u\right)+\sum _{j=1}^p{\lambda }_j^{\mathcal{R}}{\mathcal{R}}_j(\aleph )-\sum _{j=1}^p{\lambda }_j^{\mathcal{R}}{\mathcal{R}}_j\left(u\right)\hfill \\ & -\sum _{j=1}^p{\mu }_j^{\mathcal{R}}{\mathcal{R}}_j(\aleph )+\sum _{j=1}^p{\mu }_j^{\mathcal{R}}{\mathcal{R}}_j\left(u\right)-\sum _{r=1}^l{\lambda }_r^{\mathcal{S}}{\mathcal{S}}_r(\aleph )+\sum _{r=1}^l{\lambda }_r^{\mathcal{S}}{\mathcal{S}}_r\left(u\right)-\sum _{r=1}^l{\lambda }_r^{\mathcal{T}}{\mathcal{T}}_r(\aleph )\hfill \\ & +\sum _{r=1}^l{\lambda }_r^{\mathcal{T}}{\mathcal{T}}_r\left(u\right)\hfill \\ & \ge ⟨\xi +\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\xi }_r^{\mathcal{Q}}+\sum _{j=1}^p\left[{\lambda }_j^{\mathcal{R}}{\eta }_j+{\mu }_j^{\mathcal{R}}{\nu }_j\right]+\sum _{r=1}^l\left[{\lambda }_r^{\mathcal{S}}{\xi }_r^{\mathcal{S}}+{\lambda }_r^{\mathcal{T}}{\xi }_r^{\mathcal{T}}\right],\aleph -u⟩.\hfill \end{array}$$

From (4), there exists $\overline{{\xi }_r}\in co{\partial }^T\mathcal{P}\left(u\right)$, ${\overline{{\xi }_r}}^{\mathcal{Q}}\in co{\partial }^T{\mathcal{Q}}_r\left(u\right)$, $\overline{{\eta }_j}\in co{\partial }^T{\mathcal{R}}_j\left(u\right)$, $\overline{{\nu }_j}\in co{\partial }^T(-{\mathcal{R}}_j)\left(u\right)$, ${\overline{{\xi }_r}}^{\mathcal{S}}\in co{\partial }^T(-{\mathcal{S}}_r)\left(u\right)$ and ${\overline{{\xi }_r}}^{\mathcal{T}}\in co{\partial }^T(-{\mathcal{T}}_r)\left(u\right)$, such that

$$\begin{array}{c}\hfill \overline{\xi }+\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\overline{\xi }}_r^{\mathcal{Q}}+\sum _{j=1}^p\left[{\lambda }_j^{\mathcal{R}}{\overline{\eta }}_j+{\mu }_j^{\mathcal{R}}{\overline{\nu }}_j\right]+\sum _{r=1}^l\left[{\lambda }_r^{\mathcal{S}}{\overline{\xi }}_r^{\mathcal{S}}+{\lambda }_r^{\mathcal{T}}{\overline{\xi }}_r^{\mathcal{T}}\right]=0.\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill & \mathcal{P}(\aleph )-\mathcal{P}\left(u\right)\hfill \\ \hfill & +\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\mathcal{Q}}_r(\aleph )-\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\mathcal{Q}}_r\left(u\right)+\sum _{j=1}^p{\lambda }_j^{\mathcal{R}}{\mathcal{R}}_j(\aleph )-\sum _{j=1}^p{\lambda }_j^{\mathcal{R}}{\mathcal{R}}_j\left(u\right)-\sum _{j=1}^p{\mu }_j^{\mathcal{R}}{\mathcal{R}}_j(\aleph )\hfill \\ & +\sum _{j=1}^p{\mu }_j^{\mathcal{R}}{\mathcal{R}}_j\left(u\right)-\sum _{r=1}^l{\lambda }_r^{\mathcal{S}}{\mathcal{S}}_r(\aleph )+\sum _{r=1}^l{\lambda }_r^{\mathcal{S}}{\mathcal{S}}_r\left(u\right)-\sum _{r=1}^l{\lambda }_r^{\mathcal{T}}{\mathcal{T}}_r(\aleph )+\sum _{r=1}^l{\lambda }_r^{\mathcal{T}}{\mathcal{T}}_r\left(u\right)\hfill \\ & \ge 0.\hfill \end{array}$$

From the feasibility of ℵ for the problem $\left(P\right)$, ${\mathcal{Q}}_r(\aleph )\le 0$, ${\mathcal{R}}_j(\aleph )=0$, ${\mathcal{S}}_r(\aleph )\ge 0$, ${\mathcal{T}}_r(\aleph )\ge 0$, we get

$$\begin{array}{cc}\hfill & \mathcal{P}(\aleph )-\mathcal{P}\left(u\right)\hfill \\ & -\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\mathcal{Q}}_r\left(u\right)-\sum _{j=1}^p{\lambda }_j^{\mathcal{R}}{\mathcal{R}}_j\left(u\right)+\sum _{j=1}^p{\mu }_j^{\mathcal{R}}{\mathcal{R}}_j\left(u\right)+\sum _{r=1}^l{\lambda }_r^{\mathcal{S}}{\mathcal{S}}_r\left(u\right)+\sum _{r=1}^l{\lambda }_r^{\mathcal{S}}{\mathcal{T}}_r\left(u\right)\ge 0.\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill \mathcal{P}(\aleph )\ge \mathcal{P}\left(u\right)+\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\mathcal{Q}}_r\left(u\right)+\sum _{j=1}^p{\theta }_j^{\mathcal{R}}{\mathcal{R}}_j\left(u\right)-\sum _{r=1}^l\left[{\lambda }_r^{\mathcal{S}}{\mathcal{S}}_r\left(u\right)+{\lambda }_r^{\mathcal{T}}{\mathcal{T}}_r\left(u\right)\right],\end{array}$$

where ${\theta }_j^{\mathcal{R}}={\lambda }_j^{\mathcal{R}}-{\mu }_j^{\mathcal{R}}$. □

Theorem 2

(Strong Duality). For the considered primal problem $\left(P\right)$ suppose $\overline{\aleph }$ and $\mathcal{P}$ are local optimal solutions and locally Lipschitz near $\overline{\aleph }$, respectively. Suppose that $\mathcal{P},{\mathcal{Q}}_r(r\in I_{\mathcal{Q}}),\underline{+}{\mathcal{R}}_j(j=1,\dots ,p),-{\mathcal{S}}_r(r\in \Theta \cup \Omega ),-{\mathcal{T}}_r(r\in {\rm Y}\cup \Omega )$ are tangentially convex functions and are ${\partial }^T$-convex functions at $\overline{\aleph }$. If $GS-ACQ$ [15] satisfied at $\overline{\aleph }$, then ∃$\overline{\lambda }=({\overline{\lambda }}^{\mathcal{Q}},{\overline{\lambda }}^{\mathcal{R}},{\overline{\lambda }}^{\mathcal{S}},{\overline{\lambda }}^{\mathcal{T}})\in {\mathbb{R}}^{k+p+2l}$, such that $(\overline{\aleph },\overline{\lambda })$ is an optimal solution of the dual $WD\left(\overline{\aleph }\right)$ and the objective values of primal and dual are equal.

Proof. 

For the problem $\left(P\right)$, let $\overline{\aleph }$ be a locally optimal solution where the $GS$-ACQ holds at $\overline{\aleph }$. Then, by (Cor. 5.1, [15]), there exist $\overline{\lambda }=({\overline{\lambda }}^{\mathcal{Q}},{\overline{\lambda }}^{\mathcal{R}},{\overline{\lambda }}^{\mathcal{S}},{\overline{\lambda }}^{\mathcal{T}})\in {\mathbb{R}}^{k+p+2l}$ and $\overline{\mu }=({\overline{\mu }}^{\mathcal{R}},{\overline{\mu }}^{\mathcal{S}},{\overline{\mu }}^{\mathcal{T}})\in {\mathbb{R}}^{p+2l}$ such that the $GS$-stationary conditions (Def. 5.4, [15]) for the problem $\left(P\right)$ are satisfied. Specifically, there exist $\overline{\xi }\in co{\partial }^T\mathcal{P}\left(u\right)$, ${\overline{\xi }}^{\mathcal{Q}}\in co{\partial }^T{\mathcal{Q}}_r\left(u\right)$, ${\overline{\eta }}_j\in co{\partial }^T{\mathcal{R}}_j\left(u\right)$, ${\overline{\nu }}_j\in co{\partial }^T(-{\mathcal{R}}_j)\left(u\right)$, ${\overline{\xi }}^{\mathcal{S}}\in co{\partial }^T(-{\mathcal{S}}_r)\left(u\right)$, and ${\overline{\xi }}^{\mathcal{T}}\in co{\partial }^T(-{\mathcal{T}}_r)\left(u\right)$, such that

$$\begin{array}{c}\hfill \overline{\xi }+\sum _{r\in I_{\mathcal{Q}}}{\overline{{\lambda }_r}}^{\mathcal{Q}}{\overline{\xi }}_r^{\mathcal{Q}}+\sum _{j=1}^p\left[{\overline{{\lambda }_j}}^{\mathcal{R}}{\overline{\eta }}_j+{\overline{{\mu }_j}}^{\mathcal{R}}{\overline{\nu }}_j\right]+\sum _{r=1}^l\left[{\overline{{\lambda }_r}}^{\mathcal{S}}{\overline{\xi }}_r^{\mathcal{S}}+{\overline{{\lambda }_r}}^{\mathcal{T}}{\overline{\xi }}_r^{\mathcal{T}}\right]=0.\end{array}$$

$$\begin{array}{cc}\hfill & {\overline{\lambda }}_{I_{\mathcal{Q}}}^{\mathcal{Q}}\ge 0,{\overline{{\lambda }_j}}^{\mathcal{R}},{\overline{{\mu }_j}}^{\mathcal{R}}\ge 0,j=1,\dots ,p,{\overline{{\lambda }_r}}^{\mathcal{S}},{\overline{{\lambda }_r}}^{\mathcal{T}},{\overline{{\mu }_r}}^{\mathcal{S}},{\overline{{\mu }_r}}^{\mathcal{T}}\ge 0,r=1,\dots ,l,\hfill \\ & {\overline{{\lambda }_{{\rm Y}}}}^{\mathcal{S}}={\overline{{\lambda }_{\Theta }}}^{\mathcal{T}}={\overline{{\mu }_{{\rm Y}}}}^{\mathcal{S}}={\overline{{\mu }_{\Theta }}}^{\mathcal{T}}=0,\forall r\in \Omega ,{\overline{{\mu }_r}}^{\mathcal{S}}=0,{\overline{{\mu }_r}}^{\mathcal{T}}=0.\hfill \end{array}$$

Therefore, $(\overline{\aleph },\overline{\lambda })$ is feasible for the dual $WD\left(\overline{\aleph }\right)$. By Theorem 1, we have

$$\begin{array}{c}\hfill \mathcal{P}(\aleph )\ge \mathcal{P}\left(u\right)+\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\mathcal{Q}}_r\left(u\right)+\sum _{j=1}^p{\theta }_j^{\mathcal{R}}{\mathcal{R}}_j\left(u\right)-\sum _{r=1}^l\left[{\lambda }_r^{\mathcal{S}}{\mathcal{S}}_r\left(u\right)+{\lambda }_r^{\mathcal{T}}{\mathcal{T}}_r\left(u\right)\right],\end{array}$$

(11)

where ${\theta }_j^{\mathcal{R}}={\lambda }_j^{\mathcal{R}}-{\mu }_j^{\mathcal{R}}$, from the feasibility condition of $\left(P\right)$ and any feasible solution $(u,\lambda )$ for the dual $WD\left(\overline{\aleph }\right)$, that is, for $r\in I_{\mathcal{Q}}\left(\overline{\aleph }\right)$, ${\mathcal{Q}}_r\left(\overline{\aleph }\right)=0$, and ${\mathcal{R}}_j\left(\overline{\aleph }\right)=0$, ${\mathcal{S}}_r\left(\overline{\aleph }\right)=0$, $\forall r\in \Theta \cup \Omega$, ${\mathcal{T}}_r\left(\overline{\aleph }\right)=0$, $\forall r\in \Omega \cup {\rm Y}$, then, we have

$$\begin{array}{c}\hfill \mathcal{P}\left(\overline{\aleph }\right)=\mathcal{P}\left(\overline{\aleph }\right)+\sum _{r\in I_{\mathcal{Q}}}{\overline{\lambda }}_r^{\mathcal{Q}}{\mathcal{Q}}_r\left(\overline{\aleph }\right)+\sum _{j=1}^p{\overline{\theta }}_j^{\mathcal{R}}{\mathcal{R}}_j\left(\overline{\aleph }\right)-\sum _{r=1}^l\left[{\overline{\lambda }}_r^{\mathcal{S}}{\mathcal{S}}_r\left(\overline{\aleph }\right)+{\overline{\lambda }}_r^{\mathcal{T}}{\mathcal{T}}_r\left(\overline{\aleph }\right)\right],\end{array}$$

(12)

where ${\overline{{\theta }_j}}^{\mathcal{R}}={\overline{{\lambda }_j}}^{\mathcal{R}}-{\overline{{\mu }_j}}^{\mathcal{R}}.$ Using (11) and (12), we have

$$\begin{array}{cc}\hfill \mathcal{P}\left(\overline{\aleph }\right)& +\sum _{r\in I_{\mathcal{Q}}}{\overline{\lambda }}_r^{\mathcal{Q}}{\mathcal{Q}}_r\left(\overline{\aleph }\right)+\sum _{j=1}^p{\overline{\theta }}_j^{\mathcal{R}}{\mathcal{R}}_j\left(\overline{\aleph }\right)-\sum _{r=1}^l\left[{\overline{\lambda }}_r^{\mathcal{S}}{\mathcal{S}}_r\left(\overline{\aleph }\right)+{\overline{\lambda }}_r^{\mathcal{T}}{\mathcal{T}}_r\left(\overline{\aleph }\right)\right]\hfill \\ & \ge \mathcal{P}\left(u\right)+\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\mathcal{Q}}_r\left(u\right)+\sum _{j=1}^p{\theta }_j^{\mathcal{R}}{\mathcal{R}}_j\left(u\right)-\sum _{r=1}^l\left[{\lambda }_r^{\mathcal{S}}{\mathcal{S}}_r\left(u\right)+{\lambda }_r^{\mathcal{T}}{\mathcal{T}}_r\left(u\right)\right].\hfill \end{array}$$

Hence, $(\overline{\aleph },\overline{\lambda })$ is an optimal solution for the dual $WD\left(\overline{\aleph }\right)$ and the respective objective values are equal. □

Example 1.

Suppose the following nonsmooth MPEC problem $\left(P\right)$ in ${\mathbb{R}}^2$ with tangentially convex functions:

$$\begin{array}{cc}\hfill \left(P\right):\mathit{min}\mathcal{P}({\aleph }_1,{\aleph }_2)=|{\aleph }_1|+|{\aleph }_2|+{\aleph }_2^2& \\ \hfill \mathit{subject}\mathit{to}:\mathcal{Q}({\aleph }_1,{\aleph }_2)=\left|{\aleph }_2\right|\le 0& \\ \hfill \mathcal{S}({\aleph }_1,{\aleph }_2)={\aleph }_1& \ge 0,\mathcal{T}({\aleph }_1,{\aleph }_2)={\aleph }_2\ge 0,\hfill \\ \hfill \mathcal{S}({\aleph }_1,{\aleph }_2)\mathcal{T}({\aleph }_1,{\aleph }_2)& =0.\hfill \end{array}$$

Clearly, all the functions are ${\partial }^T$-convex at $\overline{\aleph }=(0,0)$. Tangential subdifferentials at $\overline{\aleph }=(0,0)$ are ${\partial }^T\mathcal{P}(0,0)=\{(-1,0),(1,0)\}$, ${\partial }^T\mathcal{Q}(0,0)=\{(0,1),(0,-1)\}$, ${\partial }^T(-\mathcal{S})(0,0)=\left\{(-1,0)\right\}$, ${\partial }^T(-\mathcal{T})(0,0)=\left\{(0,-1)\right\}$. Now, we easily prove that this problem $\left(P\right)$ satisfies all the conditions of Theorems 1 and 2.

For the considering primal problem $\left(P\right)$, we formulate Mond–Weir-type dual $\left(MWD\right)$ and discuss duality theorems using tangential subdifferentials.

$$MWD\left(\overline{\aleph }\right):\underset{u,\lambda }{max}\left\{\mathcal{P}\left(u\right)\right\}$$

subject to

$$\begin{array}{cc}\hfill & 0\in co{\partial }^T{\mathcal{P}}_r\left(u\right)+\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}co{\partial }^T{\mathcal{Q}}_r\left(u\right)+\sum _{j=1}^p\left[{\lambda }_j^{\mathcal{R}}co{\partial }^T{\mathcal{R}}_j\left(u\right)+{\mu }_j^{\mathcal{R}}co{\partial }^T(-{\mathcal{R}}_j)\left(u\right)\right]\hfill \\ & +\sum _{r=1}^l\left[{\lambda }_r^{\mathcal{S}}co{\partial }^T(-{\mathcal{S}}_r)\left(u\right)+{\lambda }_r^{\mathcal{S}}co{\partial }^T(-{\mathcal{T}}_r)\left(u\right)\right],\hfill \\ & {\mathcal{Q}}_r\left(u\right)\ge 0(r\in I_{\mathcal{Q}}),{\mathcal{R}}_r\left(u\right)=0(r=1,\dots ,p),\hfill \\ & {\mathcal{S}}_r\left(u\right)\le 0(r\in \Theta \cup \Omega ),{\mathcal{T}}_r\left(u\right)=0(r\in \Omega \cup {\rm Y}),\hfill \\ & {\lambda }_{I_{\mathcal{Q}}}^{\mathcal{S}}\ge 0,{\lambda }_j^{\mathcal{R}},{\mu }_j^{\mathcal{R}}\ge 0,j=1,\dots ,p,{\lambda }_r^{\mathcal{S}},{\lambda }_r^{\mathcal{T}},{\mu }_r^{\mathcal{S}},{\mu }_r^{\mathcal{T}}\ge 0,r=1,\dots ,l,\hfill \\ & {\lambda }_{{\rm Y}}^{\mathcal{S}}={\lambda }_{\Theta }^{\mathcal{T}}={\mu }_{{\rm Y}}^{\mathcal{S}}={\mu }_{\Theta }^{\mathcal{T}}=0,\forall r\in \Omega ,{\mu }_r^{\mathcal{S}}=0,{\mu }_r^{\mathcal{T}}=0,\hfill \end{array}$$

(13)

where

• $\lambda =({\lambda }^{\mathcal{Q}},{\lambda }^{\mathcal{R}},{\lambda }^{\mathcal{S}},{\lambda }^{\mathcal{T}})\in {\mathbb{R}}^{k+p+2l}$ and $\mu =({\mu }^{\mathcal{R}},{\mu }^{\mathcal{S}},{\mu }^{\mathcal{T}})\in {\mathbb{R}}^{p+2l}.$

Theorem 3

(Weak Duality). Let us consider the primal problem denoted as $\left(P\right)$ along with its associated dual problem $MWD\left(\overline{\aleph }\right)$. Assume that $\overline{\aleph }$ is feasible for the primal problem and $(u,\lambda )$ is feasible for the dual problem. Define the index sets $I_{\mathcal{Q}},\Theta ,\Omega ,{\rm Y}$ as appropriate to the context.

We examine the functions $\mathcal{P}$, ${\mathcal{Q}}_r$ for $r\in I_{\mathcal{Q}}$, $\underline{+}{\mathcal{R}}_j$ for $j=1,\dots ,p$, $-{\mathcal{S}}_r$ for $r\in \Theta \cup \Omega$, and $-{\mathcal{T}}_r$ for $r\in {\rm Y}\cup \Omega$ which are characterized as tangentially convex as well as ${\partial }^T$-convex at the point u.

If the union of the sets ${\Omega }_{\mu }^{\mathcal{S}}$, ${\Omega }_{\mu }^{\mathcal{T}}$, ${\Theta }_{\mu }^{+}$, and ${{\rm Y}}_{\mu }^{+}$ is empty, i.e.,

$${\Omega }_{\mu }^{\mathcal{S}}\cup {\Omega }_{\mu }^{\mathcal{T}}\cup {\Theta }_{\mu }^{+}\cup {{\rm Y}}_{\mu }^{+}=\varnothing ,$$

then for any feasible point ℵ pertaining to the problem $\left(P\right)$, the following holds:

$$\mathcal{P}(\aleph )\ge \mathcal{P}\left(u\right).$$

Proof. 

Since $\mathcal{P}$ is convex at u, then,

$$\begin{array}{c}\hfill \mathcal{P}(\aleph )-\mathcal{P}\left(u\right)\ge ⟨\xi ,\aleph -u⟩,\forall \xi \in {\partial }^T\mathcal{P}\left(u\right).\end{array}$$

(14)

Similarly, we have

$$\begin{array}{c}\hfill {\mathcal{Q}}_r(\aleph )-{\mathcal{Q}}_r\left(u\right)\ge ⟨{\xi }_r^{\mathcal{Q}},\aleph -u⟩,\forall {\xi }_r^{\mathcal{Q}}\in {\partial }^T{\mathcal{Q}}_r\left(u\right),\forall r\in I_{\mathcal{Q}},\end{array}$$

(15)

$$\begin{array}{c}\hfill {\mathcal{R}}_j(\aleph )-{\mathcal{R}}_j\left(u\right)\ge ⟨{\eta }_j,\aleph -u⟩,\forall {\eta }_j\in {\partial }^T{\mathcal{R}}_j\left(u\right),\forall j=\{1,\dots ,p\},\end{array}$$

(16)

$$\begin{array}{c}\hfill -{\mathcal{R}}_j(\aleph )+{\mathcal{R}}_j\left(u\right)\ge ⟨{\nu }_j,\aleph -u⟩,\forall {\nu }_j\in {\partial }^T(-{\mathcal{R}}_j)\left(u\right),\forall j=\{1,\dots ,p\},\end{array}$$

(17)

$$\begin{array}{c}\hfill -{\mathcal{S}}_r(\aleph )+{\mathcal{S}}_r\left(u\right)\ge ⟨{\xi }_r^{\mathcal{S}},\aleph -u⟩,\forall {\xi }_r^{\mathcal{S}}\in {\partial }^T(-{\mathcal{S}}_r)\left(u\right),\forall r\in \Theta \cup \Omega ,\end{array}$$

(18)

$$\begin{array}{c}\hfill -{\mathcal{T}}_r(\aleph )+{\mathcal{T}}_r\left(u\right)\ge ⟨{\xi }_r^{\mathcal{T}},\aleph -u⟩,\forall {\xi }_r^{\mathcal{T}}\in {\partial }^T(-{\mathcal{T}}_r)\left(u\right),\forall r\in \Omega \cup {\rm Y}.\end{array}$$

(19)

If ${\Omega }_{\mu }^{\mathcal{S}}\cup {\Omega }_{\mu }^{\mathcal{T}}\cup {\Theta }_{\mu }^{+}\cup {{\rm Y}}_{\mu }^{+}=\varnothing ,$ then multiplying (15)–(19) by ${\lambda }_r^{\mathcal{Q}}\ge 0(r\in I_{\mathcal{Q}}),{\lambda }_j^{\mathcal{R}}>0(j=1,\dots ,p),{\mu }_j^{\mathcal{R}}>0(j=1,\dots ,p),{\lambda }_r^{\mathcal{S}}>0(r\in \Theta \cup \Omega ),{\lambda }_r^{\mathcal{T}}>0(r\in {\rm Y}\cup \Omega )$, respectively, and adding (14)–(19), we get

$$\begin{array}{cc}\hfill & \mathcal{P}(\aleph )-\mathcal{P}\left(u\right)+\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\mathcal{Q}}_r(\aleph )-\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\mathcal{Q}}_r\left(u\right)+\sum _{j=1}^p{\lambda }_j^{\mathcal{R}}{\mathcal{R}}_j(\aleph )-\sum _{j=1}^p{\lambda }_j^{\mathcal{R}}{\mathcal{R}}_j\left(u\right)\hfill \\ & -\sum _{j=1}^p{\mu }_j^{\mathcal{S}}{\mathcal{S}}_j(\aleph )+\sum _{j=1}^p{\mu }_j^{\mathcal{R}}{\mathcal{R}}_j\left(u\right)-\sum _{r=1}^l{\lambda }_r^{\mathcal{S}}{\mathcal{S}}_r(\aleph )+\sum _{r=1}^l{\lambda }_r^{\mathcal{S}}{\mathcal{S}}_r\left(u\right)-\sum _{r=1}^l{\lambda }_r^{\mathcal{T}}{\mathcal{T}}_r(\aleph )+\sum _{r=1}^l{\lambda }_r^{\mathcal{T}}{\mathcal{T}}_r\left(u\right)\hfill \\ & \ge ⟨\xi +\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\xi }_r^{\mathcal{Q}}+\sum _{j=1}^p\left[{\lambda }_j^{\mathcal{R}}{\eta }_j+{\mu }_j^{\mathcal{R}}{\nu }_j\right]+\sum _{r=1}^l\left[{\lambda }_r^{\mathcal{S}}{\xi }_r^{\mathcal{S}}+{\lambda }_r^{\mathcal{T}}{\xi }_r^{\mathcal{T}}\right],\aleph -u⟩.\hfill \end{array}$$

From duality results, there exists $\overline{{\xi }_r}\in co{\partial }^T\mathcal{P}\left(u\right)$, ${\overline{{\xi }_r}}^{\mathcal{Q}}\in co{\partial }^T{\mathcal{Q}}_r\left(u\right)$, $\overline{{\eta }_j}\in co{\partial }^T{\mathcal{R}}_j\left(u\right)$, $\overline{{\nu }_j}\in co{\partial }^T(-{\mathcal{R}}_j)\left(u\right)$, ${\overline{{\xi }_r}}^{\mathcal{S}}\in co{\partial }^T(-{\mathcal{S}}_r)\left(u\right)$ and ${\overline{{\xi }_r}}^{\mathcal{T}}\in co{\partial }^T(-{\mathcal{T}}_r)\left(u\right)$, such that

$$\begin{array}{c}\hfill \overline{\xi }+\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\overline{\xi }}_r^{\mathcal{Q}}+\sum _{j=1}^p\left[{\lambda }_j^{\mathcal{R}}{\overline{\eta }}_j+{\mu }_j^{\mathcal{R}}{\overline{\nu }}_j\right]+\sum _{r=1}^l\left[{\lambda }_r^{\mathcal{S}}{\overline{\xi }}_r^{\mathcal{S}}+{\lambda }_r^{\mathcal{T}}{\overline{\xi }}_r^{\mathcal{T}}\right]=0.\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill & \mathcal{P}(\aleph )-\mathcal{P}\left(u\right)\hfill \\ \hfill & +\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\mathcal{Q}}_r(\aleph )-\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\mathcal{Q}}_r\left(u\right)+\sum _{j=1}^p{\lambda }_j^{\mathcal{R}}{\mathcal{R}}_j(\aleph )-\sum _{j=1}^p{\lambda }_j^{\mathcal{R}}{\mathcal{R}}_j\left(u\right)-\sum _{j=1}^p{\mu }_j^{\mathcal{R}}{\mathcal{R}}_j(\aleph )\hfill \\ & +\sum _{j=1}^p{\mu }_j^{\mathcal{R}}{\mathcal{R}}_j\left(u\right)-\sum _{r=1}^l{\lambda }_r^{\mathcal{S}}{\mathcal{S}}_r(\aleph )+\sum _{r=1}^l{\lambda }_r^{\mathcal{S}}{\mathcal{S}}_r\left(u\right)-\sum _{r=1}^l{\lambda }_r^{\mathcal{T}}{\mathcal{T}}_r(\aleph )+\sum _{r=1}^l{\lambda }_r^{\mathcal{T}}{\mathcal{T}}_r\left(u\right)\hfill \\ & \ge 0.\hfill \end{array}$$

Now, applying the feasibility of ℵ and u for the primal $\left(P\right)$, and $MWD\left(\overline{\aleph }\right)$, respectively, we get

$$\mathcal{P}(\aleph )\ge \mathcal{P}\left(u\right).$$

□

Corollary 1.

Consider the primal problem $\left(P\right)$, and let $\overline{\aleph }$ be a feasible solution in which all the constraint functions ${\mathcal{Q}}_r$, ${\mathcal{R}}_r$, ${\mathcal{S}}_r$, ${\mathcal{T}}_r$ are affine, with the index sets $I_{\mathcal{Q}}$, Θ, Ω, Y defined in the usual manner. Then, for any feasible solution ℵ of $\left(P\right)$ and any feasible pair $(u,\lambda )$ of the corresponding dual problem $MWD\left(\overline{\aleph }\right)$, the strict inequality

$$\mathcal{P}(\aleph )<\mathcal{P}\left(u\right),$$

cannot occur.

Example 2.

Consider the following problem in ${\mathbb{R}}^2$:

$$\begin{array}{cc}\hfill \left(P\right):\mathit{min}\mathcal{P}({\aleph }_1,{\aleph }_2)={\aleph }_1+\left|{\aleph }_2\right|& \\ \hfill \mathit{subject}\mathit{to}:\mathcal{S}({\aleph }_1,{\aleph }_2)={\aleph }_1+{\aleph }_2& \ge 0,\hfill \\ \hfill \mathcal{T}({\aleph }_1,{\aleph }_2)={\aleph }_1-2{\aleph }_2& \ge 0,\hfill \\ \hfill \mathcal{S}({\aleph }_1,{\aleph }_2)\mathcal{T}({\aleph }_1,{\aleph }_2)& =0.\hfill \end{array}$$

Here we see that the considered functions are tangentially convex. Let $\overline{\aleph }=(0,0)$ be the feasible point, then the tangential subdifferentials at (0, 0) are ${\partial }^T\mathcal{P}(0,0)=\left\{1\right\}\times [-1,1]$, ${\partial }^T\mathcal{S}(0,0)=\left\{(1,1)\right\}$, ${\partial }^T\mathcal{T}(0,0)=\left\{(1,-2)\right\}$

$$MWD\left(\overline{\aleph }\right)\underset{u,\lambda }{max}{u}_1+\left|u_2\right|$$

subject to: $\exists {\xi }_1\in co{\partial }^T(u_1+|u_2\left|\right)=\left\{1\right\}\times [-1,1]$, such that

$$\left(\begin{array}{c}0\\ 0\end{array}\right)=\left(\begin{array}{c}1\\ \xi \end{array}\right)-{\lambda }^{\mathcal{S}}\left(\begin{array}{c}1\\ 1\end{array}\right)-{\lambda }^{\mathcal{T}}\left(\begin{array}{c}1\\ -2\end{array}\right)$$

(20)

where $\xi \in [-1,1]$

$$\begin{array}{c}\hfill {\lambda }^{\mathcal{S}}(u_1+u_2)\le 0,{\lambda }^{\mathcal{T}}(u_1-2u_2)\le 0,{\lambda }^{\mathcal{S}}\ge 0,{\lambda }^{\mathcal{T}}\ge 0.\end{array}$$

(21)

From (20), we have ${\lambda }^{\mathcal{S}}+{\lambda }^{\mathcal{T}}=1$ and ${\lambda }^{\mathcal{S}}-2{\lambda }^{\mathcal{T}}=\xi$. This leads us to the expressions ${\lambda }^{\mathcal{S}}={\displaystyle \frac{\xi +2}{3}}$ and ${\lambda }^{\mathcal{T}}={\displaystyle \frac{1-\xi }{3}}$. Assuming $\overline{\aleph }=(0,0)$, the index sets $\Theta (0,0)$ and ${\rm Y}(0,0)$ are both empty, whereas $\Omega (0,0)$ is non-empty. Therefore, the conditions of Corollary 1 are satisfied. Next, if $\xi \in (-1,1)$, then we find ${\lambda }^{\mathcal{S}}>0$ and ${\lambda }^{\mathcal{T}}>0$. After resolving Equation (21), we obtain $u_1+|u_2|\le 0\le \mathcal{P}(\aleph )\le {\aleph }_1+\left|{\aleph }_2\right|$ for all feasible ℵ. Similarly, when $\overline{\aleph }=(1,-1)$, the index sets $\Omega (1,-1)$ and ${\rm Y}(1,-1)$ are empty, while $\Theta (1,-1)$ is non-empty. In all cases examined, the conditions of Corollary 1 continue to hold. Thus, we confirm that Corollary 1 is satisfied between the primal problem $\left(P\right)$ and its dual $MWD\left(\overline{\aleph }\right)$.

Theorem 4

(Strong Duality). Suppose that $\overline{\aleph }$ is a local optimal solution of the primal problem $\left(P\right)$, and let $\mathcal{P}$ P be tangentially convex and locally Lipschitz continuous at this point. Further, assume that the functions $\overline{\aleph }$: $\mathcal{P}$, ${\mathcal{Q}}_r(r\in I_{\mathcal{Q}})$, $\underline{+}{\mathcal{R}}_j(j=1,\dots ,p)$, $-{\mathcal{S}}_r(r\in \Theta \cup \Omega )$, and $-{\mathcal{T}}_r(r\in {\rm Y}\cup \Omega )$ are tangentially convex and ${\partial }^T$-convex at $\overline{\aleph }$. If the generalized standard Abadie constraint qualification (GS-ACQ) is satisfied at $\overline{\aleph }$ (see [15]), then there exists a multiplier vector $\overline{\lambda }$ such that the pair $(\overline{\aleph },\overline{\lambda })$ is an optimal solution of the dual problem $MWD\left(\overline{\aleph }\right)$, and moreover, the objective values of the primal and its dual coincide.

Proof. 

The proof follows the same steps as the proof of Theorem 2 by using Theorem 3. □

We consider the primal problem $\left(P\right)$ and its $MWD$, we establish weak and strong duality theorems under ${\partial }^T$-convexity assumptions.

Theorem 5

(Weak Duality). Suppose that $\overline{\aleph }$ and $(u,\lambda )$ are feasible for the primal problem $\left(P\right)$ and its $MWD\left(\overline{\aleph }\right)$, respectively, and the index sets $I_{\mathcal{Q}},\Theta ,\Omega ,{\rm Y}$ are defined accordingly. Let $\mathcal{P}$ be tangentially convex and ${\partial }^T$-pseudoconvex at u, ${\mathcal{Q}}_r(r\in I_{\mathcal{Q}})$, $\underline{+}{\mathcal{R}}_j(j=1,\dots ,p)$, $-{\mathcal{S}}_r(r\in \Theta \cup \Omega )$, $-{\mathcal{T}}_r(r\in {\rm Y}\cup \Omega )$ are tangentially convex and are ${\partial }^T$-quasiconvex functions at u. If ${\Omega }_{\mu }^{\mathcal{S}}\cup {\Omega }_{\mu }^{\mathcal{T}}\cup {\Theta }_{\mu }^{+}\cup {{\rm Y}}_{\mu }^{+}=\varnothing$, then for any ℵ feasible for the problem $\left(P\right)$, we have

$$\mathcal{P}(\aleph )\ge \mathcal{P}\left(u\right).$$

Proof. 

Let a feasible point ℵ, such that $\mathcal{P}(\aleph )<\mathcal{P}\left(u\right)$ then, by pseudoconvexity of $\mathcal{P}$ at u, we have

$$\begin{array}{c}\hfill ⟨\xi ,\aleph -u⟩<0,\forall \xi \in {\partial }^T\mathcal{P}\left(u\right).\end{array}$$

(22)

From (13), there exists $\overline{\xi }\in co{\partial }^T\mathcal{P}\left(u\right)$, ${\overline{{\xi }_r}}^{\mathcal{Q}}\in co{\partial }^T{\mathcal{Q}}_r\left(u\right)$, $\overline{{\eta }_j}\in co{\partial }^T{\mathcal{R}}_j\left(u\right)$, $\overline{{\nu }_j}\in co{\partial }^T(-{\mathcal{R}}_j)\left(u\right)$, ${\overline{{\xi }_r}}^{\mathcal{S}}\in co{\partial }^T(-{\mathcal{S}}_r)\left(u\right)$ and ${\overline{{\xi }_r}}^{\mathcal{T}}\in co{\partial }^T(-{\mathcal{T}}_r)\left(u\right)$, such that

$$\begin{array}{c}\hfill -\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\overline{\xi }}_r^{\mathcal{Q}}-\sum _{j=1}^p\left[{\lambda }_j^{\mathcal{R}}{\overline{\eta }}_j+{\mu }_j^{\mathcal{R}}{\overline{\nu }}_j\right]-\sum _{\Theta \cup \Omega }{\lambda }_r^{\mathcal{S}}{\overline{\xi }}_r^{\mathcal{S}}-\sum _{\Omega \cup {\rm Y}}{\lambda }_r^{\mathcal{T}}{\overline{\xi }}_r^{\mathcal{T}}\in {\partial }^T\mathcal{P}\left(u\right).\end{array}$$

(23)

By (22), we get

$$\begin{array}{c}\hfill ⟨\left(\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\overline{\xi }}_r^{\mathcal{Q}}+\sum _{j=1}^p\left[{\lambda }_j^{\mathcal{R}}{\overline{\eta }}_j+{\mu }_j^{\mathcal{R}}{\overline{\nu }}_j\right]+\sum _{\Theta \cup \Omega }{\lambda }_r^{\mathcal{S}}{\overline{\xi }}_r^{\mathcal{S}}+\sum _{\Omega \cup {\rm Y}}{\lambda }_r^{\mathcal{T}}{\overline{\xi }}_r^{\mathcal{T}}\right),\aleph -u⟩>0.\end{array}$$

(24)

For each $r\in I_{\mathcal{Q}},{\mathcal{Q}}_r(\aleph )\le 0\le {\mathcal{Q}}_r\left(u\right)$. Hence, by ${\partial }^T$-quasiconvexity, we have

$$\begin{array}{c}\hfill ⟨\xi ,\aleph -u⟩\le 0,\forall \xi \in {\partial }^T{\mathcal{Q}}_r\left(u\right),\forall r\in I_{\mathcal{Q}}.\end{array}$$

(25)

Similarly, we have

$$\begin{array}{c}\hfill ⟨{\eta }_j,\aleph -u⟩\le 0,\forall {\eta }_j\in {\partial }^T{\mathcal{R}}_j\left(u\right),\forall j=\{1,\dots ,p\}.\end{array}$$

(26)

Now, for any feasible point u of the dual $MWD\left(\overline{\aleph }\right)$, and for each $j,-{\mathcal{R}}_j\left(u\right)={\mathcal{R}}_j(\aleph )=0$, also, $-{\mathcal{S}}_r(\aleph )\le -{\mathcal{S}}_r\left(u\right)$, $\forall r\in \Theta \cup \Omega$, and $-{\mathcal{T}}_r(\aleph )\le -{\mathcal{T}}_r\left(u\right),\forall r\in {\rm Y}\cup \Omega$. By ${\partial }^T$-quasiconvexity, we have

$$\begin{array}{cc}\hfill & ⟨{\nu }_j,\aleph -u⟩,\forall {\nu }_j\in {\partial }^T(-{\mathcal{R}}_j)\left(u\right),\forall j=\{1,\dots ,p\},\hfill \end{array}$$

(27)

$$\begin{array}{cc}& ⟨{\xi }_r^{\mathcal{S}},\aleph -u⟩,\forall {\xi }_r^{\mathcal{S}}\in {\partial }^T(-{\mathcal{S}}_r)\left(u\right),\forall r\in \Theta \cup \Omega ,\hfill \end{array}$$

(28)

$$\begin{array}{cc}& ⟨{\xi }_r^{\mathcal{T}},\aleph -u⟩,\forall {\xi }_r^{\mathcal{T}}\in {\partial }^T(-{\mathcal{T}}_r)\left(u\right),\forall r\in \Omega \cup \mathrm{Y}.\hfill \end{array}$$

(29)

From Equations (25)–(29), we obtain

$$\begin{array}{cc}\hfill & ⟨{\overline{\xi }}_r^{\mathcal{Q}},\aleph -u⟩\le 0(r\in I_{\mathcal{Q}}),⟨{\overline{\eta }}_j,\aleph -u⟩\le 0,⟨{\overline{\nu }}_j,\aleph -u⟩\le 0(r=1,\dots ,p),\hfill \\ & ⟨{\overline{\xi }}_r^{\mathcal{S}},\aleph -u⟩\le 0,\forall r\in \Theta \cup \Omega ,⟨{\overline{\xi }}_r^{\mathcal{T}},\aleph -u⟩\le 0,\forall r\in {\rm Y}\cup \Omega .\hfill \end{array}$$

Since ${\Omega }_{\mu }^{\mathcal{S}}\cup {\Omega }_{\mu }^{\mathcal{T}}\cup {\Theta }_{\mu }^{+}\cup {{\rm Y}}_{\mu }^{+}=\varnothing$, we have

$$\begin{array}{cc}\hfill & ⟨\sum _{\Theta \cup \Omega }{\lambda }_r^{\mathcal{S}}{\overline{\xi }}_r^{\mathcal{S}},\aleph -u⟩\le 0,⟨\sum _{\Omega \cup {\rm Y}}{\lambda }_r^{\mathcal{T}}{\overline{\xi }}_r^{\mathcal{T}},\aleph -u⟩\le 0,\hfill \\ & ⟨\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\overline{\xi }}_r^{\mathcal{Q}},\aleph -u⟩\le 0,⟨\sum _{j=1}^p\left[{\lambda }_j^{\mathcal{R}}{\overline{\eta }}_j+{\mu }_j^{\mathcal{R}}{\overline{\nu }}_j\right],\aleph -u⟩\le 0.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill ⟨\left(\sum _{r\in I_{\mathcal{Q}}}{\lambda }_r^{\mathcal{Q}}{\overline{\xi }}_r^{\mathcal{Q}}+\sum _{j=1}^p\left[{\lambda }_j^{\mathcal{R}}{\overline{\eta }}_j+{\mu }_j^{\mathcal{R}}{\overline{\nu }}_j\right]+\sum _{\Theta \cup \Omega }{\lambda }_r^{\mathcal{S}}{\overline{\xi }}_r^{\mathcal{S}}+\sum _{\Omega \cup {\rm Y}}{\lambda }_r^{\mathcal{T}}{\overline{\xi }}_r^{\mathcal{T}}\right),\aleph -u⟩\le 0,\end{array}$$

which contradicts (24). Hence, $\mathcal{P}(\aleph )\ge \mathcal{P}\left(u\right)$. □

Theorem 6

(Strong Duality). Suppose that $\overline{\aleph }$ is a local optimal solution of the problem $\left(P\right)$, and let $\mathcal{P}$ be tangentially convex and locally Lipschitz at that point. Additionally, assume that $\mathcal{P}$ is ${\partial }^T$-pseudoconvex at $\overline{\aleph }$, and that ${\mathcal{Q}}_r(r\in I_{\mathcal{Q}})$, $\underline{+}{\mathcal{R}}_j(j=1,\dots ,p)$, $-{\mathcal{S}}_r(r\in \Theta \cup \Omega )$, and $-{\mathcal{T}}_r(r\in {\rm Y}\cup \Omega )$ are all tangentially convex and ${\partial }^T$-quasiconvex functions at $\overline{\aleph }$. If the generalized standard Abadie constraint qualification (GS-ACQ) is satisfied at $\overline{\aleph }$, as discussed in [15], then there exists a multiplier vector $\overline{\lambda }$ such that the pair $(\overline{\aleph },\overline{\lambda })$ constitutes an optimal solution of the dual problem $MWD\left(\overline{\aleph }\right)$. Moreover, under this condition, the objective values of the primal and the dual coincide.

Proof. 

The proof follows the same steps as the proof of Theorem 4 by using Theorem 5. □

4. Conclusions

In this work, we considered mathematical programs with equilibrium constraints (MPECs) under the assumption that both the objective and constraint functions are tangentially convex. We developed and analyzed the Wolfe-type and Mond–Weir-type dual formulations of the primal problem (P) by employing the framework of tangential subdifferentials. Furthermore, we established weak and strong duality theorems that link the primal problem (P) with its corresponding duals, under conditions of convexity and generalized convexity. In particular, the results were derived using the notions of ${\partial }^T$-convexity, ${\partial }^T$-pseudoconvexity and ${\partial }^T$-quasiconvexity.