Steering exact penalty DCA for nonsmooth DC optimization problems with equality and inequality constraints

TL;DR

This paper introduces a novel DCA-based method with an adaptive steering exact penalty approach for solving complex nonsmooth DC optimization problems with constraints, demonstrating improved convergence and practical effectiveness.

Contribution

It develops a new penalty updating strategy within DCA for nonsmooth constrained DC problems, with detailed convergence analysis and practical applications.

Findings

Convergence of the proposed method is rigorously established.

The method effectively solves nonsmooth constrained optimization problems.

Numerical experiments show improved performance over existing approaches.

Abstract

We propose and study a version of the DCA (Difference-of-Convex functions Algorithm) using the $\ell_1$ penalty function for solving nonsmooth DC optimization problems with nonsmooth DC equality and inequality constraints. The method employs an adaptive penalty updating strategy to improve its performance. This strategy is based on the so-called steering exact penalty methodology and relies on solving some auxiliary convex subproblems to determine a suitable value of the penalty parameter. We present a detailed convergence analysis of the method and illustrate its practical performance by applying the method to two nonsmooth discrete optimal control problem.