Optimization on directionally convex sets

Directional convexity generalizes the concept of classical convexity. We investigate OC-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 OC-convexity being infinite. Given a compact OC-convex set A, maximize a linear form L subject to A. We prove that there exists an OC-extreme solution of the problem. We introduce the notion of OC-quasiconvex function. Ii is shown that if O is finite then the constrained maximum of an OC-quasiconvex function on the set A is attained at an OC-extreme point of A. We show that the OC-convex hull of a finite point set represents the union of a finite set of polytopes in case of the direction set being finite.