Lopsided convergence: an extension and its quantification

Abstract

Much of the development of lopsided convergence for bifunctions defined on product spaces was in response to motivating applications. A wider class of applications requires an extension that would allow for arbitrary domains, not only product spaces. This leads to an extension of the definition and its implications that include the convergence of solutions and optimal values of a broad class of minsup problems. In the process we relax the definition of lopsided convergence even for the classical situation of product spaces. We now capture applications in optimization under stochastic ambiguity, Generalized Nash games, and many others. We also introduce the lop-distance between bifunctions, which leads to the first quantification of lopsided convergence. This quantification facilitates the study of convergence rates of methods for solving a series of problems including minsup problems, Generalized Nash games, and various equilibrium problems.