Regularity and sufficient optimality conditions for some classes of mathematical programming problems (Q749864)

Let f be a functional defined on a real Banach space X, C a nonempty closed convex subset of X, F a map from X into a real Banach space Y and K a closed convex cone in Y with vertex at the origin. We derive sufficient optimality conditions for the mathematical programming problem \[ \text{ minimize } f(x)\text{ subject to } F(x)\in K,\quad x\in C. \] The paper is organized as follows. In Section 2, using the generalized open mapping theorem of \textit{J. Zowe} and \textit{S. Kurcyusz} [Appl. Math. Optimization 5, 49-62 (1979; Zbl 0401.90104)], we show that under their regularity assumptions, the feasible set can be approximated by the linearizing cone. In Section 3, some first and second-order sufficient optimality conditions are obtained. Section 4 is devoted to the discussion of a case studied by \textit{V. M. Alekseev}, \textit{V. M. Tikhomirov} and \textit{S. V. Fomin} [``Optimal control'' (1979; Zbl 0516.49002); for a review of the 1987 English translation see Zbl 0689.49001] and \textit{E. S. Levitin}, \textit{A. A. Milyutin} and \textit{N. P. Osmolovskij} [Sov. Math., Dokl. 14, 853-857 (1973); translation from Dokl. Akad. Nauk. SSSR 210, 1022-1025 (1973; Zbl 0292.49009)].