Duality in reverse convex optimization

A duality theorem for the general problem of minimizing an extended real-valued convex function on a locally convex linear space under a reverse convex constraint is considered. In the particular case of the distance to a reverse convex subset in a normed linear space, we recover as a corollary a duality theorem due to C. Franchetti and I. Singer [Boll. Un. Mat. Ital. B (5), 17 (1980), pp. 33-43] similar to the one known for the distance to a convex subset. The general theorem also contains the duality principle of Toland-Singer for D.C. optimization.