The Boosted Double-Proximal Subgradient Algorithm for Nonconvex Optimization

TL;DR

The paper introduces BDSA, a new algorithm for nonconvex optimization that combines subgradient and proximal methods with line-search, extending existing schemes and demonstrating effectiveness on complex test functions and practical problems.

Contribution

It presents BDSA, a novel splitting algorithm for structured nonconvex nonsmooth problems, with convergence analysis and practical applications.

Findings

BDSA effectively escapes non-optimal critical points.

Convergence is established under Kurdyka--Lojasiewicz property.

Demonstrated success on test functions and real-world problems.

Abstract

In this paper we introduce the Boosted Double-proximal Subgradient Algorithm (BDSA), a novel splitting algorithm designed to address general structured nonsmooth and nonconvex mathematical programs expressed as sums and differences of composite functions. BDSA exploits the combined nature of subgradients from the data and proximal steps, and integrates a line-search procedure to enhance its performance. While BDSA encompasses existing schemes proposed in the literature, it extends its applicability to more diverse problem domains. We establish the convergence of BDSA under the Kurdyka--Lojasiewicz property and provide an analysis of its convergence rate. To evaluate the effectiveness of BDSA, we introduce a novel family of challenging test functions with an abundance of critical points. We conduct comparative evaluations demonstrating its ability to effectively escape non-optimal critical points. Additionally, we present two practical applications of BDSA for testing its efficacy, namely, a constrained minimum-sum-of-squares clustering problem and a nonconvex generalization of Heron's problem.