Control and Cybernetics 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 α(·)-paraconvex. Let (X, ‖.‖) be a real Banach space. Let f(x) be a real valued strongly α(·)-paraconvex function defined on an open convex subset Ω ⊂ X , i.e. f ( tx+(1− t)y ≤ tf(x)+(1− t)f(y)+C min[t, (1− t)]α(‖x−y‖). Then there is a set of the first Baire category AF ⊂ Ω such that the function f(·) is Fréchet differentiable at every point x0 ∈ Ω \AF .