Continuous approximate solutions in parametric optimization

Abstract

Alma'mct. The question of the existence of approximate solutions in parametric optimization is considered. Most results show that (under hypotheses) if a certain optimization problem has an approximate solution x0 for a value Po of a parameter, then an approximate solution x = b(p) can be found for p in P, with b continuous, b(po) = Xo, and any two such bs are homotopic. Some topological methods (use of fibrations) are used to weaken the usual 'convex' hypotheses of such results. An equisemicontinuity condition (relative to a constraint) is introduced to allow some noncompactness. The results are applied to get approximate Nash equilibrium results for games with some nonconvexity in the strategy sets.AMS subject classifications (1980). 90C31, 90D10, 54C60.