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

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 optimisation 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.

Keywords:

• DC function
• DC optimisation
• nonsmooth optimisation
• DCA
• exact penalty method

AMS Classifications:

• 90C26
• 90C30
• 90C56
• 65K05

Acknowledgments

The author wishes to express his sincere gratitude to the anonymous referees for carefully reading the paper and providing valuable comments that helped to significantly improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was performed in IPME RAS and supported by the Russian Science Foundation [grant number 20-71-10032].

Notes on contributors

M. V. Dolgopolik

M. V. Dolgopolik received a specialist (M.Sc.) degree in Applied Mathematics from St. Petersburg State University in 2012, Candidate of Sciences (Ph.D.) degree in Discrete Mathematics and Mathematical Cybernetics in 2015, and Doctor of Sciences (Habilitation) degree in Real, Complex, and Functional Analysis in 2022 from the same university. Since 2015, he is a senior research fellow at the Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, Saint Petersburg, Russia. His current research interests include nonsmooth and DC optimization, nonsmooth and variational analysis, computational geometry, calculus of variations, optimal control, and control of PDE.