Categorical properties of solvability for one class of minimization problems
Let \((E,\|\cdot\|)\) be a real Banach space with closed unit ball \(B_E\) and \(B\) a closed subset of \(E\) such that, for some \(\beta >0,\) \(\,B_E\subset B\subset \beta B_E\). Then the Minkowski functional (or the gauge) of the set \(B\) satisfies the inequalities \(\beta^{-1}\|\cdot\|\leq\mu_B(\cdot)\leq \|\cdot\|.\) One says that \(B\) has the Kadets-Klee (KK) property if for every sequence \((x_n)_{n\geq 0}\) in \(E\) with \(\mu_B(x_n)=1\), \(n=0,1,\dots\), \(x_n \rightarrow x_0\) weakly implies \(x_n \rightarrow x_0\) strongly. The main result of the paper (Theorem 1) asserts that if the Banach space \(E\) is reflexive and \(B\) has the property (KK), then for every nonempty closed subset \(X\) of \(E\) and every bounded from below lsc function \(f\) on \(E\), there exists a \(G_\delta\) set \(G\) dense in \(E\setminus X \) such that, for every \(y\in G\), the problem \(f(x)+\mu_B(x-y)\rightarrow \inf_{x\in X}\) has a solution. It is shown that the result does not hold if the space \(E\) is not reflexive or if \(E\) is reflexive but \(B\) does not have the property (KK). Applications are given to linear systems with a vector performance criterion. This kind of approximation, called generalized approximation, was initiated by \textit{F. S. De Blasi} and \textit{J. Myjak} [J. Approximation Theory 94, No. 1, 54--72 (1998; Zbl 0912.41015)], who considered the case of a uniformly convex set \(B\). \textit{C. Li} [J. Approximation Theory 107, No. 1, 96--108 (2000; Zbl 1014.41015)] proved a generic existence result for \(B\) strictly convex and (KK) and a weakly compact set \(X\); see also the survey paper by the reviewer [Abstr. Appl. Anal. 2005, No. 3, 259--285 (2005; Zbl 1100.46005)].
scientific article; zbMATH DE number 6362542
scientific article