A convergence theory for saddle functions

Abstract

We develop a convergence theory called epi/hypo-convergence, for bivariate functions that essentially implies the convergence of their saddle points. We study the properties of this limiting process in particular. We characterize the limit functions associated to any collection of bivariate functions and obtain a compactness theorem for the space of saddle functions. Even when restricted to the univariate case, the results generalize those known for epi-convergence. In particular, we show that the analysis of the convergence process via Yosida approximates must not be restricted to the convex case.