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Generic well-posedness of optimization problems in topological spaces

Published online by Cambridge University Press:  26 February 2010

M. M. Čoban
Affiliation:
Moldavian SSR, 90 Odesskaia Str. kv 50, 278 000 Tiraspol, U.S.S.R.
P. S. Kenderov
Affiliation:
Institute of Mathematics, Bulgarian Academy of Sciences, “Acad. G. Bonchev” Str, Block 8, 1113 Sofia, Bulgaria.
J. P. Revalski
Affiliation:
Institute of Mathematics, Bulgarian Academy of Sciences, “Acad. G. Bonchev” Str, Block 8, 1113 Sofia, Bulgaria.

Abstract

Let X be a completely regular Hausdorff topological space and let C(X) (the set of all real-valued bounded and continuous in X functions) be endowed with the sup-norm. Let ßX, as usual, denotes the Stone-Čech compactification of X. We give a characterization of those X for which the set

contains a dense -subset of C(X). These are just the spaces X which contain a dense Čech complete subspace. We call such spaces almost Čech complete. We also prove that X contains a dense completely metrizable subspace, if, and only if, C(X) contains a dense -subset of functions which determine Tykhonov well-posed optimization problems over X. For a compact Hausdorff topological space X the latter result was proved by Čoban and Kenderov [CK1.CK2]. Relations between the well-posedness and Gâteaux and Fréchet differentiability of convex functionals in C(X) are investigated. In particular it is shown that the sup-norm in C(X) is Frechet differentiable at the points of a dense -subset of C(X), if, and only if, the set of isolated points of X is dense in X. Conditions and examples are given when the set of points of Gateaux differentiability of the sup-norm in C(X) is a dense and Baire subspace of C(X) but does not contain a dense -subset of C(X).

Type
Research Article
Copyright
Copyright © University College London 1989

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