Porosity of perturbed optimization problems in Banach spaces
Let \(X\) be a Banach space and \(Z\) a nonempty closed subset of \(X\). Let \(J:Z\to\mathbb R\) be a lower semicontinuous function bounded from below. This paper is concerned with the perturbed optimization problem \(\inf_{z\in Z}\{J(z)+\|x-z\|\}\), denoted by \((x,J)\)-inf for \(x\in X\). In the case when \(X\) is compactly fully 2-convex, it is proved in the present paper that the set of all points \(x\) in \(X\) for which there does not exist \(z_0\in Z\) such that \(J(z_0)+ \|x-z_0\|= \inf_{z\in Z}\{J(z0+ \|x-z\|\}\) is a \(\sigma\)-porous set in \(X\). Furthermore, if \(X\) is assumed additionally to be compactly locally uniformly convex, we verify that the set of all points \(x\in X\setminus Z^0\) such that the problem \((x,J)\)-inf fails to be approximately compact, is a \(\sigma\)-porous set in \(X\setminus Z^0\), where \(Z^0\) denotes the set of all \(z\in Z\) such that \(z\in P_Z(z)\). Moreover, a counterexample to which some results of \textit{R. X. Ni} [J. Math. Anal. Appl. 302, No. 2, 417--424 (2005; Zbl 1061.49002)] fail is provided.
scientific article; zbMATH DE number 5072981
scientific article