A dual view of equilibrium problems

Giancarlo Bigi, Marco Castellani and Gábor Kassay


This paper introduces and studies a dual problem associated to a generalized equilibrium problem (GEP). Some duality results had already been presented for equilibrium problems but actually an optimization problem had been introduced as a dual, relying on a gap function. On the contrary, the approach of this paper allows to introduce a dual equilibrium problem with no need of a gap function and without formulating it as an optimization problem. We introduce a dual generalized equilibrium problem (DGEP) in such a way that each solution of (GEP) provides a solution of (DGEP) and vice versa. Moreover, we prove that the solutions of (GEP) and (DGEP) are strictly related to the saddle points of the Lagrangian function associated to these problems. We construct a family of parametric minimization problems (primal problems) which serves as a solvability test for (GEP). The Lagrangian representation for the Fenchel duality framework provides the corresponding family of maximization problems (dual problems) with the same set of parameters. It turns out that, in some circumstances, the solution set of (GEP) coincides with the union of the solution sets of the primal problems, while the solution set of (DGEP) coincides with the union of the solution sets of the dual problems. We also pay special attention to the case in which the equilibrium bifunction is Frèchet differentiable with respect to its second variable. In this situation our results can be formulated in a simpler way. The last section is devoted to the application of our approach to relevant particular cases of equilibrium problems. It turns out that known duality results for optimization and the duality theorem for variational inequalities due to Mosco can be recovered.

If you are interested in this paper, feel free to contact me.

My Home Page