Criteria for Epi/Hypo Convergence of Finite-Valued Bifunctions

Abstract

Epi/hypo convergence of finite-valued bivariate functions defined on the product of two subsets, with some connections to lopsided convergence, is considered. Namely, we deal with three full characterizations of this convergence: by epi/hypo convergence of the corresponding proper bifunctions, by explicit formulae of the lower and upper members of the intervals of the limits, and by the bicontinuity of the partial Legendre-Fenchel transform (i.e., the (extended) epi/hypo convergence of bifunctions is characterized by the epi-convergence of their convex parents). We emphasize that epi/hypo limits are not unique and form an entire equivalence class.

Approximations of Optimization-Related Problems in Terms of Variational Convergence

“…(ii) Limits of an e/h-convergent sequence are not unique. The limits form a class of bifunctions, called an e/h-equivalence class, see, e.g., [7]. However, as we will see below, fortunately almost all variational properties are the same for all limit bifunctions in an equivalence class.…”

“…However, as we will see below, fortunately almost all variational properties are the same for all limit bifunctions in an equivalence class. (iii) In [7], characterizations of e/h-convergence and lop-convergence of finitevalued bifunctions were established. In particular, [7, Theorem 3] asserted the equivalence of the e/h-convergence of a sequence of finite-valued bifunctions and the e/h-convergence of the corresponding proper extended-real-valued bifunctions.…”

“…Here and later on, we focus on epi/hypo convergence, since it is symmetric and appropriate for investigations of a saddle point, the basic object one is often interested in, when dealing with bivariate functions involved in optimization-related problems. We show that any cluster point of a sequence of (approximate) saddle points of bivariate functions which epi/hypo converge is an (approximate) saddle point of any limit bivariate function (epi/hypo limits are not unique, see, e.g., [7]), without any additional assumption. Then, we define notions of ancillary tightness and (full) tightness to ensure the Painlevé-Kuratowski convergence of the whole set of saddle points of epi/hypo convergent bivariate functions.…”

“…In Example 2 of[7], it was computed that all the bifunctionsK a (x, y) = y x if (x, y) ∈ [0, 1] 2 \ {(0, 0)}, a if (x, y) = (0, 0)for any a ∈ [0, 1], are e/h-limits of the sequence {K ν }, i.e., they form the e/h-equivalence class. It is easy to check that K ν e/ h → K a fully tightly.…”

Approximations of Variational Problems in Terms of Variational Convergence

“…In [7,8] lopside convergence of finite-valued bifunctions defined on rectangles (i.e., product sets) was introduced and studied with applications in approximation/stability of variational problems. In [9,10] epi/hypo convergence of finite-valued bifunctions defined on rectangles was developed for the finite-dimensional case. For unifunctions the books [11,12] are prominent comprehensive references.…”