Summary: 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 J. Morgan
and M. Romaniello
[JIPAM, J. Inequal. Pure Appl. Math. 4, No. 2, Paper No. 28, 7 p., electronic only (2003; Zbl 1082.65544
)], which extend those introduced by A. Auslender and M. Teboulle
[SIAM J. Optim., 10, No. 4, 1097–1115 (2000; Zbl 0996.49005
)] for a variational inequality problem.