Optimality Functions and Lopsided Convergence.

摘要

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 apply it to nonlinear programming. This results in a convergence result for a primal interior point method without constraint qualifications or convexity assumptions. 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.