An equivalence relation between optimization problems connected with the well-posedness

Abstract

Let X be a Hausdorff topological space and 23 be a family of real-valued functions defined in X which are bounded from below. Taking an arbitrary f~ 3 we want to minimize f on X, i.e., to find X,EX such that f(xo) = inf{f(x): x E X}. This problem will be referred to as the minimization problem (X,f). Varying f in B we obtain a set of minimization problems. We are interested in those functions from 23 for which the corresponding minimization problem has a unique solution and, moreover, some continuous dependence of this solution on the data of the problem exists. Such problems are called well-posed (the precise definition is given below). Suppose b is endowed with a uniform metric under which it is a complete metric space (two important cases are: (i) 8 consists of lower semicontinuous functions; (ii) 23 consists of continuous functions). In this case the following question is of some interest: Does the set {f~ 23: (X,f) is well-posed) contain a dense and G,-subset of 8 (and from the point of view of the Baire category is considered to be a “big” subset of d)? In a different setting (including both the unconstrained and constrained case) the answer to this question is positive and is given in [l-7, l&12, 14173. A possible reservation, one may have regarding such results, is mentioned by Beer in [ 11. Following Kenderov [ 10, 111, he considers the next set of constrained minimization problems Cp = { (A,f): 0 #A c X, A is compact, f: X+ R, f is continuous and bounded}, X is Tech complete. It is observed in [l] that the reason for the topological “bigness” of the set {Mf) E ‘p: (Af) is well-posed} could be the fact that in Cp there exist different couples giving the same minimization problem. Indeed, let (A,f), (B, g) E Cp, A = B, f # g, but the restricti0ns.f 1 A and g 1 B coincide. Then, as