Paraconvex analysis

In the theory of optimization an essential role is played by the differentiability of convex functions. In this paper we shall try to extend the results concerning differentiability to a larger class of functions called strongly alpha(.)-paraconvex. Let (X, (|.||) be a real Banach space. Let f(x) be a real valued strongly alpha(.)-paraconvex function denned on an open convex subset Omega [is a subset is] X, i.e. f(tx+(1-t)y)