Iterative algorithms for equilibrium problems.

*(English)* Zbl 1176.90640
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 |

90C33 | Complementarity and equilibrium problems; variational inequalities (finite dimensions) |

49J40 | Variational methods including variational inequalities |

49M37 | Methods of nonlinear programming type in calculus of variations |

65K05 | Mathematical programming (numerical methods) |