Solvability theorems involving inf-convex functions
The function \(f : X_0 \to R \cup \{\infty\}\) is inf-convex if there exists a family of functions \((f_\alpha)_{\alpha \in \Delta} \subset \Gamma (X_0)\) such that \(f = \inf_{\alpha \in \Delta} f_\alpha\), where \(X_0\) is a nonempty convex subset of a locally convex Hausdorff vector space \(X\). Similarly one defines min-convex functions (the infimum is attained at every \(x \in X_0)\), inf-SL and min-SL functions (SL comes from sublinear). The main result of the paper is Theorem 3.1 which gives necessary and sufficient conditions for having: \(x \in K\), \(g(x) \in S \Rightarrow f(x) \geq 0\), where \(K\) and \(S\) are closed convex cones, while \(f\) and \(\lambda g\) \((\lambda \in \Lambda \subset S^*)\) are inf- SL functions. By homogenization, one extends this result to inf-convex functions. The results are then applied to Lagrangian duality for infinite-dimensional programming problems involving min-convex functions.
scientific article; zbMATH DE number 801003