Inexact proximal DC Newton-type method for nonconvex composite functions

TL;DR

This paper introduces an inexact proximal DC Newton-type method for nonconvex composite optimization, combining inexact Newton steps with proximal algorithms, and demonstrates its convergence and efficiency through numerical experiments.

Contribution

It proposes a novel inexact proximal DC Newton-type algorithm with convergence guarantees and efficient computation of scaled proximal mappings.

Findings

The method converges globally.

It outperforms existing algorithms in experiments.

A semi-smooth Newton method efficiently computes proximal mappings.

Abstract

We consider a class of difference-of-convex (DC) optimization problems where the objective function is the sum of a smooth function and a possible nonsmooth DC function. The application of proximal DC algorithms to address this problem class is well-known. In this paper, we combine a proximal DC algorithm with an inexact proximal Newton-type method to propose an inexact proximal DC Newton-type method. We demonstrate global convergence properties of the proposed method. In addition, we give a memoryless quasi-Newton matrix for scaled proximal mappings and consider a two-dimensional system of semi-smooth equations that arise in calculating scaled proximal mappings. To efficiently obtain the scaled proximal mappings, we adopt a semi-smooth Newton method to inexactly solve the system. Finally, we present some numerical experiments to investigate the efficiency of the proposed method, showing that the proposed method outperforms existing methods.