Duality gap of the conic convex constrained optimization problems in normed spaces
In this interesting paper the authors extend the results of \textit{T. Champion} [Duality gap in convex programming, Math. Program. 99, No.~3 (A), 487--498 (2004; Zbl 1084.90032)] to the problem \(\min f(x)\) s.t. \(x\in C\), \(h(x)\in-S\), where \(X,Y\) are normed spaces, \(C\subset X\) is a closed convex set, \(S\subset Y\) is a closed convex cone with nonempty interior, \(f:X\to \mathbb{R}\cup\{+\infty\}\) is a proper and lower semicontinuous functional, and \(h:X\to Y\) is an \(S\)-convex mapping. For the study of the preceding problem it is considered the condition \({\mathcal D}(y)\) which means that \(\bigcap_{z>y}\overline{\{h\leq z\}\cap C+\ker x^*}=\overline{\{h\leq y\}\cap C+\ker x^*}\) for every \(x^*\in X^*\), where \(z>y\) \((z\geq y)\) means that \(z-y\in\operatorname{int}S\) \((z-y\in S)\) and \(\{h\leq y\}= \{x\in X\mid h(x)\leq y\}.\) The condition \({\mathcal D}(y)\) is an analog of a condition used by T. Champion.
scientific article; zbMATH DE number 5547235
scientific article