Iterative algorithms for equilibrium problems. (English) Zbl 1176.90640

Iusem, Alfredo N.; Sosa, Wilfredo

Optimization 52, No. 3, 301-316 (2003).

Summary: We consider equilibrium problems in the framework of the formulation proposed by Blum and Oettli, which includes variational inequalities, Nash equilibria in noncooperative games, and vector optimization problems, for instance, as particular cases. We show that such problems are particular instances of convex feasibility problems with infinitely many convex sets, but with additional structure, so that projection algorithms for convex feasibility can be modified in order to improve their convergence properties, mainly achieving global convergence without either compactness or coercivity assumptions. We present a sequential projections algorithm with an approximately most violated constraint control strategy, and two variants where exact orthogonal projections are replaced by approximate ones, using separating hyperplanes generated by subgradients. We include full convergence analysis of these algorithms.

MSC:

90C47	Minimax problems in mathematical programming
90C33	Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
49J40	Variational inequalities
49M37	Numerical methods based on nonlinear programming
65K05	Numerical mathematical programming methods

Keywords:

Equilibrium Problems; Convex Feasibility Problems; Projection Methods