Preservation of Convergence of Convex Sets and Functions in Finite Dimensions

Abstract

We study a convergence notion which has particular relevance for convex analysis and lends itself quite naturally to successive approximation schemes in a variety of areas.Motivated particularly by problems in optimization subject to constraints, we develop technical tools necessary for systematic use of this convergence in finite-dimensional settings.Simple conditions are established under which this convergence for sequences of sets, functions and subdifferentials is preserved under various basic operations, including, for example, those of addition and infimal convolution in the case of functions. Introduction.A certain very natural notion of convergence for sequences of convex sets, functions and related operators has received increasing attention since the mid-1960's.Such work has been motivated by efforts toward successive approximation schemes in a wide variety of areas, including statistics, variational inequalities, approximation theory, stochastic optimization, control theory, and mathematical programming.Especially noteworthy features of this convergence notion, as applied to convex functions, are that it is preserved under two of the chief operations of convex analysis: the Fenchel transform, and subdifferentiation.In the context of functions, the notion differs in subtle yet important ways from ordinary pointwise convergence.In this paper we analyze how this convergence behaves with respect to various basic constructions involving convex sets, functions and subdifferentials.Our aim is to develop the technical tools necessary for a systematic application of this convergence notion to constrained optimization situations.Elsewhere [4], [10] we make essential use of the tools developed here.For example, in [4] we develop the details of a broad scheme for approximating a given dual pair of optimization problems as the limit of a sequence of such pairs of problems.Since the source of the sequence of approximating problems is left unspecified, the framework permits applications of both a theoretical and a numerical analytical nature.The plan of the paper is as follows.In §2 we define the convergence notion for sets, functions and operators, and then the two basic results concerning preservation under conjugacy and subdifferentiation are recalled.In §3 we treat preservation of convergence of sequences of convex sets under the operations of addition