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Duality for generalized equilibrium problem. (English) Zbl 1152.90648
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.

MSC:
90C46Optimality conditions, duality
46A20Duality theory of topological linear spaces
49N15Duality theory (optimization)