A note on the convergence of nonconvex line search

TL;DR

This paper analyzes the convergence properties of line search methods for nonconvex algorithms within the Kurdyka-Lojasiewicz framework, establishing weak and global convergence results under specific conditions, with an application to sparse least squares.

Contribution

It provides new convergence results for nonconvex line search algorithms based on Kurdyka-Lojasiewicz theory, including an application to L0-regularized least squares.

Findings

Weak convergence of line search in general nonconvex algorithms.

Global convergence under Kurdyka-Lojasiewicz property and additional assumptions.

Application to L0-regularized least squares minimization.

Abstract

In this note, we consider the line search for a class of abstract nonconvex algorithm which have been deeply studied in the Kurdyka-Lojasiewicz theory. We provide a weak convergence result of the line search in general. When the objective function satisfies the Kurdyka-Lojasiewicz property and some certain assumption, a global convergence result can be derived. An application is presented for the L0-regularized least square minimization in the end of the paper.