Minimal convex majorants of functions and Demyanov--Rubinov super(sub)differentials

TL;DR

This paper characterizes functions on normed spaces that can be expressed as the lower envelope of convex majorants, extending Demyanov-Rubinov theory to broader classes and introducing a generalized subdifferential.

Contribution

It generalizes Demyanov-Rubinov characterization to non-positively homogeneous functions and introduces a new Demyanov-Rubinov subdifferential for nonsmooth functions.

Findings

Extended characterization of functions as lower envelopes of convex majorants.

Introduced a new Demyanov-Rubinov subdifferential generalizing known subdifferentials.

Applied subdifferentials to extremal problems.

Abstract

The primary goal of the paper is to establish characteristic properties of (extended) real-valued functions defined on normed vector spaces that admit the representation as the lower envelope of their minimal (with respect to pointwise ordering) convex majorants. The results presented in the paper generalize and extend the well-known Demyanov-Rubinov characterization of upper semicontinuous positively homogeneous functions as the lower envelope of exhaustive families of continuous sublinear functions to more larger classes of (not necessarily positively homogeneous) functions defined on arbitrary normed spaces. As applications of the above results, we introduce, for nonsmooth functions, a new notion of the Demyanov-Rubinov subdifferential at a given point, and show that it generalizes a number of known notions of subdifferentiability, in particular, the Fenchel-Moreau subdifferential of convex functions and the Dini-Hadamard (directional) subdifferential of directionally differentiable functions. Some applications of Demyanov-Rubinov subdifferentials to extremal problems are considered.