On radially epidifferentiable functions and regularity conditions in nonconvex optimization

TL;DR

This paper introduces the radial epiderivative for nonconvex functions, establishing its properties and relationships with other derivatives, and uses it to derive new regularity conditions and optimality criteria in nonconvex optimization.

Contribution

It presents a new definition and properties of the radial epiderivative, linking it to existing derivatives and applying it to regularity and optimality conditions in nonconvex optimization.

Findings

Established relationships between radial epiderivative and classical derivatives

Derived necessary and sufficient conditions for global optimality

Provided explicit formulas for weak subgradients

Abstract

In this paper we study the radial epiderivative notion for nonconvex functions, which extends the (classical) directional derivative concept. The paper presents new definition and new properties for this notion and establishes relationships between the radial epiderivative, the Clarke's directional derivative, the Rockafellar's subderivative and the directional derivative. The radial epiderivative notion is used to establish new regularity conditions without convexity conditions. The paper analyzes necessary and sufficient conditions for global optimums in nonconvex optimization via the generalized derivatives studied in this paper. We establish a necessary and sufficient condition for a descent direction for radially epidifferentiable nonconvex functions. The paper presents explicit formulations for computing the weak subgradients in terms of the radial epiderivatives and vice versa, which are very important from point of view of developing solution methods for finding global optimums in nonsmooth and nonconvex optimization. All the properties and theorems presented in this paper, are illustrated and interpreted on examples.