Optimization on Directionally Convex Sets

Directional convexity generalizes the concept of classical convexity. We investigate aC-convexity generated by the intersections of C-semispaces that efficiently approximates directional convexity. We consider the following optimization problem in case of the direction set of aC-convexity being infinite. Given a compact aC-convex set A, maximize a linear form L subject to A. We prove that there exists an aC-extreme solution of the problem. A Krein-Milman type theorem has been proved for aC-convexity. We show that the aC-convex hull of a finite point set represents the union of a finite set of polytopes in case of the direction set being finite.