Mean Value Theorem for Functions Possessing First Order Convex Approximations. Applications in Optimization Theory

Abstract

A mean value theorem for functions possessing first order convex approximations in the sense of Ioffe [6] is derived. It comprises two known results for convex and locally Lipschitzian functions as particular cases. This theorem is used in order to obtain a sufficient condition for a function defined on the Cartesian product of two topological vector spaces to possess a first order convex approximation. Some applications in optimization theory are also given.