transactions of the american mathematical society Volume 327, Number 2, October 1991 CONVEXOPTIMIZATION AND THE EPI-DISTANCETOPOLOGY GERALD BEERAND ROBERTO LUCCHETTI Abstract. Let T{X) denote the proper, lower semicontinuous, convex func- tions on a Banach space X , equipped with the completely metrizable topology t of uniform convergence of distance functions on bounded sets. A function / in T(X) is called well-posed provided it has a unique minimizer, and each minimizing sequence converges to this minimizer. We show that well-posedness of / e T(X) is the minimal condition that guarantees strong convergence of approximate minima of r-approximating functions to the minimum of /. Moreover, we show that most functions in (T(X), Taw ) are well-posed, and that this fails if T(X) is topologized by the weaker topology of Mosco convergence, whenever X is infinite dimensional. Applications to metric projections are also given, including a fundamental characterization of approximative compactness. 1. Introduction Let fë(X) (resp. ^B(X)) be the closed (resp. closed and bounded) nonempty convex sets in a normed linear space X. For over a half-century the basic topology on %?B(X) has been the well-known Hausdorff metric topology [15]. How should this topology be extended to ^(X) ? The generally recognized [2, 37] successful solution in finite dimensions is the completely metrizable Fell topology,generated by all sets of the form V~ = {A e 'ê'(X) : A n V ¿ 0} where V is open in X, and (A:c)+ = {A e &(X): A c Kc} where K is a compact subset of X. Convergence of a sequence (An) to A in this topology in finite dimensions is equivalent to classical Kuratowski convergence of sets [27, §29]; alternatively, it is equivalent to the pointwise convergence of the associated sequence of distance functions (d(-, An)) to d(-, A) (see, e.g., [11, 20]). Certainly one of the most important features of the Fell topology on ^(X) is its stability with respect to duality, as established by Wijsman [41], expressed by the continuity of the polar map A -> A0, or in the case of proper lower Received by the editors April 20, 1989 and, in revised form, September 5, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 49B50, 26B25; Secondary 54B20,41A50. Key words and phrases. Convex optimization, convex function, well-posed minimization prob- lem, epi-distance topology, Mosco convergence, metric projection, approximative compactness, Chebyshev set. Research of the second author was partially supported by C.N.R. © 1991 American Mathematical Society 0002-9947/91 $1.00+ $.25 per page 795 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use