On the convergence analysis of DCA

TL;DR

This paper introduces a unified proof framework for analyzing the convergence of the Difference-of-Convex (DCA) algorithm, applicable to various DC programming problems, and investigates convergence rates under specific mathematical conditions.

Contribution

It provides a general and clean proof framework for convergence analysis of DCA, applicable to standard and constrained DC programs, and explores convergence rates under Lojasiewicz and Kurdyka-Lojasiewicz conditions.

Findings

Established global convergence of DCA sequences.

Analyzed convergence rates for objective function values and iterates.

Provided conditions under which DCA converges reliably.

Abstract

In this paper, we propose a clean and general proof framework to establish the convergence analysis of the Difference-of-Convex (DC) programming algorithm (DCA) for both standard DC program and convex constrained DC program. We first discuss suitable assumptions for the well-definiteness of DCA. Then, we focus on the convergence analysis of DCA, in particular, the global convergence of the sequence $\{x^k\}$ generated by DCA under the Lojasiewicz subgradient inequality and the Kurdyka-Lojasiewicz property respectively. Moreover, the convergence rate for the sequences $\{f(x^k)\}$ and $\{\|x^k-x^*\|\}$ are also investigated. We hope that the proof framework presented in this article will be a useful tool to conveniently establish the convergence analysis for many variants of DCA and new DCA-type algorithms.