Xi-Yin Zheng, Solvability of convex optimization problems on a *-continuous closed convex set

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DOI: 10.23952/jano.4.2022.1.09
Volume 4, Issue 1, 1 April 2022, Pages 119-128

Abstract. Given a closed convex set A in a Banach space X, motivated by the continuity of A [D. Gale, V. Klee, Continuous convex sets, Math. Scand. 7 (1959), 379-391], this paper introduces and studies the $*$-continuity of A. Without the reflexivity assumption on X, we prove that the $*$-continuity of a closed convex set A implies that, for every continuous linear (or convex) function $f:X\rightarrow\mathbb{R}$ bounded below on A, the corresponding optimization problem $\inf_{x\in A}f(x)$ is weakly well-posed solvable, and that A has the attainable separation property if A is assumed to have a nonempty interior in addition.