A Refined Inertial DC Algorithm for DC Programming

TL;DR

This paper introduces a refined inertial DC algorithm (RInDCA) with larger inertial step-size for DC programming, demonstrating improved convergence speed and solution quality through theoretical analysis and numerical experiments.

Contribution

The paper proposes a new refined inertial DC algorithm with enlarged step-size, providing convergence analysis and empirical evidence of enhanced performance.

Findings

Larger inertial step-size accelerates convergence.

RInDCA converges to a critical point under certain conditions.

Numerical results show improved efficiency in matrix copositivity and image denoising.

Abstract

In this paper we consider the difference-of-convex (DC) programming problems, whose objective function is the difference of two convex functions. The classical DC Algorithm (DCA) is well-known for solving this kind of problems, which generally returns a critical point. Recently, an inertial DC algorithm (InDCA) equipped with heavy-ball inertial-force procedure was proposed in de Oliveira et al. (Set-Valued and Variational Analysis 27(4):895--919, 2019), which potentially helps to improve both the convergence speed and the solution quality. Based on InDCA, we propose a refined inertial DC algorithm (RInDCA) equipped with enlarged inertial step-size compared with InDCA. Empirically, larger step-size accelerates the convergence. We demonstrate the subsequential convergence of our refined version to a critical point. In addition, by assuming the Kurdyka-{\L}ojasiewicz (KL) property of the objective function, we establish the sequential convergence of RInDCA. Numerical simulations on checking copositivity of matrices and image denoising problem show the benefit of larger step-size.