A dual view of equilibrium problems
Giancarlo Bigi, Marco Castellani and Gábor Kassay
Summary
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.
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