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.
We present necessary and sufficient conditions for a monotone bifunction to be maximally monotone, based on a recent characterization of maximally monotone operators. These conditions state the existence of solutions to equilibrium problems obtained by perturbing the defining bifunction in a suitable way.
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