Convergence of selections with applications in optimization

Abstract

We consider the problem of finding an easily implemented tie-breaking rule for a convergent set-valued algorithm, i.e., a sequence of compact, non-empty subsets of a metric space converging in the Hausdorff metric. Our tie-breaking rule is determined by nearest-point selections defined by "uniqueness" points in the space, i.e., points having a unique best approximation in the limit set of the convergent algorithm. Convergence of the algorithm is shown to be equivalent to convergence of all such nearest-point selections. Under reasonable additional hypotheses, all points in the metric space have the uniqueness property. Consequently, all points yield convergent nearest-point selections, i.e., tie-breaking rules, for a convergent algorithm.We then show how to apply these results to approximate solutions for the following types of problems: infinite systems of inequalities, semi-infinite mathematical programming, non-convex optimization, and infinite horizon optimization.