Mean value theorems and sufficient optimality conditions for nonsmooth functions (Q1076962)

Let X be a locally convex space. Several variants of the mean value theorem for functions \(f: X\to [-\infty,\infty]\) have been proved by \textit{J. B. Hiriart-Urruty} [Numer. Funct. Anal. Optimization 2, 1-30 (1980; Zbl 0462.49032)]. In the paper under review, the first order convex approximation of f introduced by \textit{A. D. Ioffe} [SIAM J. Control Optimization 17, 245-250 (1979; Zbl 0417.49027)] is modified to be applicable to the case when both f and its approximation may take on \(+\infty\) and -\(\infty\). Then variants of the mean value theorems are obtained in terms of subdifferentials of these modified approximations. An application to the problem of minimization of non-differentiable not necessarily convex functions on a convex subset of X is presented.