Duality for nonsmooth mathematical programming problems with equilibrium constraints

In this paper, we consider the mathematical programs with equilibrium constraints (MPECs) in Banach space. The objective function and functions in the constraint part are assumed to be lower semicontinuous. We study the Wolfe-type dual problem for the MPEC under the convexity assumption. A Mond-Weir-type dual problem is also formulated and studied for the MPEC under convexity and generalized convexity assumptions. Conditions for weak duality theorems are given to relate the MPEC and two dual programs in Banach space, respectively. Also conditions for strong duality theorems are established in an Asplund space.


Introduction
Luo et al. [] presented a comprehensive study of MPEC. Flegel and Kanzow [] obtained short and elementary proof of the optimality conditions for MPEC using the standard Fritz-John conditions. Further, Flegel and Kanzow [] introduced a new Abadie-type constraint qualification and a new Slater-type constraint qualification for the MPEC and proved that new Slater-type CQ implies new Abadie-type CQ. Ye [] considered MPEC and introduced various stationary conditions and established that it is sufficient for being globally or locally optimal under some generalized convexity assumption and obtained new constraint qualifications.
Outrata et al. [] derived necessary optimality conditions for those MPECs which can be treated by the implicit programming approach and proposed a solution method based on the bundle technique of nonsmooth optimization. Flegel et al. [] considered optimization problems with a disjunctive structure of the feasible set and obtained optimality conditions for disjunctive programs with application to MPEC using Guignard-type constraint qualifications. Movahedian and Nobakhtian [] introduced nonsmooth strong stationarity, M-stationarity and generalized the Abadie and Guignard-type constraint qualifications for nonsmooth MPEC. Movahedian where X is a Banach space, f : X → R is a lower semi-continuous (lsc) function, g : X → R k , h : X → R p , G : X → R l and H : X → R l are functions with lsc components. The use of equilibrium constraints in modeling process engineering problems is a relatively new and exciting field of research; see Raghunathan and Biegler []. Hydroeconomic river basin models (HERBM) based on mathematical programming are conventionally formulated as explicit aggregate optimization problems with a single, aggregate objective function. Britz et al. [] proposed a new solution format for hydroeconomic river basin models, based on a multiobjective optimization problem with equilibrium constraints, which allowed, inter alia, to express spatial externalities resulting from asymmetric access to water use. Wolfe [] formulated a dual program for a nonlinear programming problem. Motivated by a specific problem, namely the mathematical description of the rotating heavy chain, Toland [, ] introduced the notion of duality and established the duality theory for nonconvex optimization problems. Rockafellar [, ] studied fundamental duality theory for convex programs using a conjugate function and established a generalized version of the Fenchelís duality theorem. In the last four decades there has been an extensive interest in the duality theory of nonlinear programming problems; see Mangasarian [] and Mishra and Giorgi [].
To the best of our knowledge, the dual problem to a nonsmooth MPEC has not been given in the literature as yet.
In this paper, we introduce Wolfe-type and Mond-Weir-type dual programs to the nonsmooth MPEC. We have established weak and strong duality theorems relating the nonsmooth MPEC and the two dual programs. The paper is organized as follows: in Section , we give some preliminaries, definitions, and results. In Section , we derive weak and strong duality theorems relating to the nonsmooth MPEC and the two dual models under convexity and generalized convexity assumptions.

Preliminaries
In this section, we give some notations, basic definitions, and preliminary results, which will be used later in the paper.
The Clarke-Rockafellar subdifferential of f is defined by lim sup Given a feasible vectorz for the MPEC, we define the following index sets: The set β is known as a degenerate set. If β is empty, the vectorz is said to satisfy the strict complementarity condition. Movahedian such that the following conditions hold: The following definition of the no nonzero abnormal multiplier constraint qualification for MPEC is taken from Definition . in Movahedian and Nobakhtian [].

Duality
In this section, we formulate and study a Wolfe-type dual problem for the MPEC under the convexity assumption. A Mond-Weir-type dual problem is also formulated and studied for the MPEC under convexity and generalized convexity assumptions. The Wolfe-type dual problem is formulated as follows: Theorem . (Weak duality) Letz be feasible for MPEC where X is a Banach space, (u, λ) feasible for WDMPEC(z), and index sets I g , α, β, γ defined accordingly. Suppose that f , g i are convex at u and radially noncon- are directionally Lipschitzian, convex at u, and radially nonconstant. If α -∪ γ -∪ β -G ∪ β -H = φ, then, for any z feasible for the MPEC, we have Proof Let z be any feasible point for MPEC. Then we have g i (z) ≤ , ∀i ∈ I g and h i (z) = , i = , , . . . , p.
Since f is convex at u, Similarly, we have , respectively, and adding (.)-(.), we get Now, using the feasibility of z for MPEC, that is, Hence, This completes the proof.
The following corollary is a direct consequence of Theorem ..

Corollary . Letz be feasible for
Analogously, we have the following result for Asplund spaces.
Theorem . (Weak duality) Letz be feasible for MPEC where X is an Asplund space, (u, λ) feasible for WDMPEC(z) and index sets I g , α, β, γ defined accordingly. Suppose that Proof The proof follows the lines of the proof of Theorem ..

Theorem . (Strong duality)
Assumez is a locally optimal solution of MPEC where X is an Asplund space, such that NNAMCQ is satisfied atz and the index sets I g , α, β, γ are defined accordingly. Let f , satisfy the assumption of the Theorem .. Then there existsλ, such that (z,λ) is an optimal solution of WDMPEC(z) and the respective objective values are equal.
Example . Consider the following MPEC in R  : Now, we formulate Wolfe-type dual problem WDMPEC(z) for MPEC(): If β is non-empty, then either If we take the pointz = (, ) from the feasible region, then the index sets α(, ) and γ (, ) are empty sets, but β := β(, ) is non-empty. Also, from solving a constraint equation in the feasible region of WDMPEC(, ), we get λ G = ξ η and λ H = ξ η -u  , where η = . Since β is non-empty, we consider a β + , β + G , β + H to decide the feasible region of WDMPEC(, ). It is clear that the assumptions of Theorem . are satisfied, so Theorem . holds between MPEC() and WDMPEC(, ).
It is clear thatz = (, ) is the optimal solution of MPEC() and NNAMCQ is satisfied atz. Hence, the assumptions of the Theorem . are satisfied. Then, by Theorem ., there existsλ such that (z,λ) is an optimal solution of WDMPEC(, ) and the respective values are equal.
Also, we can see that the NNAMCQ is satisfied atz. Then by Theorem . there exists λ = (λ G ,λ H ) such that (z,λ) is an optimal solution of MWDMPEC(, ) and the optimal values are equal. Now, we establish weak and strong duality theorems for the MPEC and its Mond-Weirtype dual problem under generalized convexity assumptions.