Generic well‐posedness of optimization problems in topological spaces

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 left{f ∈ C(X): the extension of on βX attains its maximum on βXonly at point of X} contains a dense scriptGδ‐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 scriptGδ‐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 scriptGδ‐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 scriptGδ‐subset of C(X).

Fragmentability of sequences of set-valued mappings with applications to variational principles

“…Further, we show how (one part of) the generic variational principle ofČoban, Kenderov and Revalski ([ČK,ČKR1]) (in a slightly stronger form given in [DR]) follows also by Theorem 3.2.…”

“…Actually, in [ČK,ČKR1] it is proved (with f ≡ 0) that the property of X to contain a dense completely metrizable subspace characterizes the fact that the well-posed problems in C b (X) form a residual subset of C b (X). A strengthening of the above theorem (as well as of the Deville-Godefroy-Zizler principle) to get a stronger (than the first Baire category) property of smallness of the complement of the well-posed problems, the so-called σ-porosity, was done in [DR].…”

“…The above generic variational principle has been used for the study of the differentiability of the sup-norm in C b (X) (see [ČK,ČKR1]). It is also closely related to the existence of (special) winning strategies for one of the players in the BanachMazur game ( [ČKR3,KR]).…”

“…In [ČK,ČKR1,ČKR2], where the case f ≡ 0 and Z = C b (X)= {the continuous bounded functions in X}, equipped with the uniform norm g ∞ := sup{|g(x)| : x ∈ X}, was considered, an approach was proposed via the solution mapping M : C b (X) → X assigning to each g ∈ C b (X) the set of minimum points to g. This mapping is minimal and due to the fact that it is densely nonempty, the corresponding generic principle was deduced via the existence of generically defined selections of M ( [ČKR2]). Unfortunately, the latter approach cannot be used directly in the case of the Stegall principle, neither in the Deville-GodefroyZizler nor in the Ioffe-Zaslavski principle, because in these principles one does not know a priori (neither is it easily seen) whether the corresponding solution mapping is densely nonempty.…”

The Banach-Mazur game and generic existence of solutions to optimization problems

“…The following fact is proved for compact spaces X in [CK1,CK2] and for an arbitrary X in [CKR,Theorem 3.5]: The set T contains a dense and G#-subset of C(X) iff the space X contains a dense and completely metrizable subspace. A family y of subsets of X is called a net in X if for every x £ X and every open U c X, with x e U, there exists 77 e y such that x £ H c U.…”