Some further duality theorems for optimization problems with reverse convex constraint sets (Q1206958)

Given an arbitrary set \(F\), \(h: F\to\overline R=[-\infty,\infty]\) a function, a subset \(\Omega\) of a locally convex space \(X\) such that \(X\backslash\Omega\) is a convex set, and a mapping \(u: F\to X\), the author presents some duality theorems for the (global, scalar) primal infimization problem \(\alpha=\inf h(\{y\in F\mid u(y)\in\Omega\})\) and the dual problem \(\beta=\inf\lambda(W)\), where \(W\subset X^*\backslash\{0\}\) and \(\lambda(w)=\inf h(\{y\in F\mid wu(y)\geq \sup w(X\backslash \Omega)\})\), or \(\lambda(w)=\inf h(\{y\in F\mid wu(y)=\sup w(X\backslash \Omega)\})\) \((w\in W)\). An extension to the case when \(X\) is an arbitrary set and \(W\subset R^ X\backslash\{0\}\) is given.