Optimality Functions and Lopsided Convergence

Abstract

Optimality functions pioneered by E. Polak characterize stationary points, quantify the degree with which a point fails to be stationary, and play central roles in algorithm development. For optimization problems requiring approximations, optimality functions can be used to ensure consistency in approximations, with the consequence that optimal and stationary points of the approximate problems indeed are approximately optimal and stationary for an original problem. In this paper, we review the framework and illustrate its application to nonlinear programming and other areas. Moreover, we introduce lopsided convergence of bifunctions on metric spaces and show that this notion of convergence is instrumental in establishing consistency of approximations. Lopsided convergence also leads to further characterizations of stationary points under perturbations and approximations.Keywords epi-Convergence · Lopsided convergence · Consistent approximations · Optimality functions · Optimality conditionsWe dedicate this paper to our long-time friend, colleague, collaborator, and advisor Elijah (Lucien) Polak in honor of his outstanding career and to celebrate his 85th birthday. We wish him fair weather and following snow conditions. Communicated by

Approximations and solution estimates in optimization

“…A strengthening of epi-convergence ensures the convergence of minima and approximation of minimizers (see for example [28]). …”

Variational Theory for Optimization under Stochastic Ambiguity

“…The notion originated with [2] and later was modified and extended in [15,16,34,33]. We show that lopsided convergence can be used to establish the existence of solutions of optimization problems with stochastic ambiguity as well as to prove the convergence of solutions of approximate problems to those of an original problem under mild assumptions.…”