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

Abstract

Abstract.The existence of a winning strategy in the well-known BanachMazur game in a completely regular topological space X is proved to be equivalent to the generic existence of solutions of optimization problems generated by continuous functions in X .

Examples of HNN-extensions with nontrivial quasi-kernels

“…Then there exists a Σ g -Cauchy sequence {x n } n≥0 that does not have any convergent subsequence, x n+1 = x n for all n and the set M consisting of all points in the sequence is a discrete set, see Remark 3.4. Consider the lower semicontinuous f : X → R + with dom f = M and inf X f = 0 defined by (8). Note that f (x k ) − f (x m ) = m−1 i=k g(x i+1 , x i ) whenever m > k ≥ 0, and in particular, f (x n ) − f (x n+1 ) = g(x n+1 , x n ) for all n ≥ 0.…”

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

“…It is also closely related to the existence of (special) winning strategies for one of the players in the BanachMazur game ( [ČKR3,KR]). …”

Porosity of ill-posed problems

“…In fact, when f ≡ 0, the property that T contains a dense and G δ -subset of C(X) characterizes the existence of a dense completely metrizable subspace of X ( [CK,CKR1]). These results are of interest not only in optimization but also in geometry of Banach spaces, because of the fact that the well-posedness of (X, |g|), g ∈ C(X), g = 0, is equivalent to the Gâteaux differentiability of the sup-norm ||·|| ∞ at g, and also in topology in connection with topological games ( [KR1,CKR2]). For a survey on other generic properties related to well-posedness, see [KR2].…”