A nonsmooth nonconvex descent algorithm

TL;DR

This paper introduces a new descent algorithm for nonsmooth, nonconvex functions that combines set-valued gradient approximation with Armijo step control, ensuring convergence to critical points and demonstrating effectiveness on benchmark problems.

Contribution

The paper proposes a novel descent algorithm for nonsmooth nonconvex functions that integrates set-valued gradient approximation with Armijo step control, with proven convergence properties.

Findings

Algorithm converges to critical points of the function.

Effective on various benchmark problems, outperforming some existing methods.

Provides theoretical justification and practical validation.

Abstract

The paper presents a new descent algorithm for locally Lipschitz continuous functions $f:X\to\mathbb{R}$. The selection of a descent direction at some iteration point $x$ combines an approximation of the set-valued gradient of $f$ on a suitable neighborhood of $x$ (recently introduced by Mankau & Schuricht) with an Armijo type step control. The algorithm is analytically justified and it is shown that accumulation points of iteration points are critical points of $f$. Finally the algorithm is tested for numerous benchmark problems and the results are compared with simulations found in the literature.