Existence and Stability in Optimization Problems

Abstract

A central question in optimization is to know whether an optimal solution exists. Associated with this question is the stability problem. A classical result states that a lower semicontinuous function attains its minimum over a compact set. The compactness hypothesis is often violated in the context of extremum problems, and thus the need for weaker and more realistic assumptions. This chapter develops several fundamental concepts and tools revolving around asymptotic functions to derive existence and stability results for general and convex minimization problems.