Mean value theorems and sufficient optimality conditions for nonsmooth functions

Abstract

In the last few years, the mean value theorem for various classes of nondifferentiable functions has been the subject of many investigations. In particular, Hiriart-Urruty [4, Sect. II] proved some variants of this theorem for extended-real-valued functions defined on locally convex spaces. In this paper we present two other theorems which are, to some extent, generalizations of the above-mentioned results. More precisely, we introduce the notion of an upper convex approximation for a function (being a modification of the first-order convex approximation of Ioffe [ 51) and formulate the mean value theorems in terms of subdifferentials of these approximations. We first consider the class of functions f’: X-t R (where X is a locally convex space) such that the restriction offto the closed line segment [a, 61 is finite and lower semicontinuous. Our result for this case (Theorem 4.2) has a weaker form than the classical mean value theorem: the difference f(h) -f(a) is expressed by means of the subdifferential of an upper convex approximation for f at a point CE [a, b]. If f is discontinuous, c may be equal to a or h. However, if the restriction ,fl [a, h] is continuous, we obtain a stronger result (Theorem 4.3) stating that c belongs to the open line segment ]a, h[. This theorem is a generalization of Theorem 1 of [9] and, in part, of Theorem 2 of [4]. Finally, we obtain, as a corollary from Theorem 4.3, a sufficient condition for a strict local minimum of a continuous function on a convex subset of X (Theorem 5.1). This theorem extends the result of [lo] to the case of functions which may not satisfy the Lipschitz condition. In the paper we make use of some notions and theorems of convex analysis which can be found in Chapter 6 of [6].