Duality for generalized equilibrium problem

Abstract

We introduce a generalized equilibrium problem (GEP) that allow us to develop a robust dual scheme for this problem, based on the theory of conjugate functions. We obtain a unified dual analysis for interesting problems. Indeed, the Lagrangian duality for convex optimization is a particular case of our dual problem. We establish necessary and sufficient optimality conditions for GEP that become a well-known theorem given by Mosco and the dual results obtained by Morgan and Romaniello, which extend those introduced by Auslender and Teboulle for a variational inequality problem.

Farkas-Type Results for Vector-Valued Functions with Applications

“…Such a model covers the special case when F is a continuous linear operator from X to X , and H is a proper, lsc and convex function considered in [11], [17]. In this case, the problem (VVI) collapses to the ordinary variational inequality problem Find x 2 C such that F (x)(z x) 0 for all z 2 C: For a …xed x 2 C, we consider the vector optimization problem (VOP(F; H; x)) WMin fF ( x)(z x) + H(z) H( x) : z 2 Cg :…”

A note on duality of generalized equilibrium problem

“…To avert such a situation duality for equilibrium (see [8]) and generalized equilibrium problems have been proposed (see [1,6]). Martinez-Legaz and Sosa [8] established a dual formulation for equilibrium problem using the classical Fenchel conjugation, generalizing the classical convex duality theory for optimization problems.…”

Strong Duality and Optimality Conditions for Generalized Equilibrium Problems

“…The schemes proposed in that paper are extensions of a classical duality theory for variational inequalities. In spirit of convex optimization, duality results and optimality conditions have been obtained for equilibrium problems by Martínez-Legaz and Sosa [12] when = 0 and by Jacinto and Scheimberg [13] when is convex, which extended the classical convex duality results. Recently, the authors in [5] considered the generalized equilibrium problems in the case where is a DC function.…”