Session: Global Optimization I Duality for

Session: Global Optimization I
Duality for Generalized Equilibrium Problem
Flávia Morgana de O. Jacinto
[email protected]
1
Susana Scheimberg 2
[email protected]
1
Depto de Matemática, ICE/UFAM, 69077-000, Manaus, AM
PESC, COPPE/UFRJ, CP68511, Rio de Janeiro, RJ – Brazil
Research supported by PICDT/CAPES
2
Depto de Ciências da Computação, IM/UFRJ
PESC, COPPE/UFRJ, CP68511, Rio de Janeiro, RJ – Brazil
Research supported by CNPq, FAPERJ e PRONEX--Optimization.
Abstract
In this work, we introduce the following Generalized Equilibrium Problem, (GEP), in a real
topological Hausdorff vector space. Our new formulation (GEP) generalizes previous equilibrium
schemes and provides an unified framework for several problems. Actually, it extends the formulations
given by Blum and Oettli [1], Flores-Bazán [2] and Martínez-Legaz and Sosa [3]. Therefore, (GEP)
formulation includes a wide number of mathematical problems, like convex optimization, variational
inequality, fixed point, Nash equilibria, complementarity.
Moreover, (GEP) formulation covers problems that are not included in the previous equilibrium schemes,
for example, the variational inequality problem considered by Mosco in [5], also called mixed variational
inequality and the quasi-variational proble m considered by Morgan and Romaniello in [4]. The main
purpose of this work is to present a conjugate duality that generalizes the usual dual concept. We
introduce a dual scheme of (GEP) and our initial arguments are similar to those given by Martínez-Legaz
and Sosa in [3], but we use a different strategy to construct the dual objective function. We show that our
scheme maintains classical dual characteristics. The dual results of a nonlinear program are obtained as a
particular case of our dual analysis. Using our characterization of primal-dual solutions we derive the dual
problem of variational inequality obtained by Mosco [5] and the dual results considered by Morgan and
Romaniello in [4] for a generalized quasi-variational problem.
References:
[1] E, BLUM and W. OETTLI, “From optimization and variational inequalities to equilibrium problems”, The Math. Student, 63
(1994), 123--145.
[2] F. FLORES-BAZÁN, “Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case”,
SIAM J. Optim. , 11 (3)(2000), 675--690.
[3] J. MARTÍNEZ-LEGAZ and W. SOSA, “Duality for equilibrium problems”, Preprint (2003).
[4] J. MORGAN and M. ROMANIELLO, “Generalized quasi-variational inequalities and duality”, J. of Ineq. in Pure and
Applied Math. 4 (18) (2003).
[5] U. MOSCO, “Dual variational inequalities”, J. of Math. Anal. and Applic. 40 (1972), 202--206.