A closedness condition and its applications to DC programs with convex constraints

This paper concerns a closedness condition called (CC), requiring a convex function and a convex constrained system. This type of condition has played an important role in the study of convex optimization problems. Our aim is to establish several characterizations of this condition and to apply them to study minimizing problems involving a DC function under a cone-convex constraint and a set constraint. First, we establish several so-called “Toland-Fenchel-Lagrange” duality theorems. As consequences of these results, various versions of generalized Farkas lemmas in dual forms and optimality conditions for DC problem are obtained. A class of DC programs with semi-definite constraints is examined as an illustration. Most of these results are established under the (CC) and our paper serves as a link between several corresponding known ones published recently for DC programs and for convex programs.