A dynamic multiple proximal bundle algorithm for locally Lipschitz multiobjective optimization problems

Abstract

We propose a method for nonsmooth nonconvex multiobjective optimization problems that is based on a bundle-type approach. The proposed method employs an extension of the redistributed proximal bundle algorithm and uses an augmented improvement function to handle different objectives. In which, at each iteration, a common piecewise linear model is used to approximate the augmented improvement function. Contrary to many existing multiobjective optimization methods, this algorithm works directly with objective functions, without using any kind of a priori chosen parameters or employing any scalarization. Under appropriate assumptions, we discuss convergence to points which satisfy a necessary condition for Pareto optimality. We provide numerical results for a set of nonsmooth convex and nonconvex multiobjective optimization problems in the form of tables, figures and performance profiles. The numerical results confirm the superiority of the proposed algorithm in the considered test problems compared to other multiobjective solvers, in the computational effort necessary to compute weakly Pareto stationary points.

KEYWORDS:

• Proximal bundle algorithm
• nonsmooth optimization
• nonconvex optimization
• multiobjective optimization

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 49J52
• 90C26
• 90C29

Acknowledgments

The research of the first author was in part supported by a grant from IPM (No. 1403900313). The authors would like to thank anonymous referees for their comments that helped to improve the quality of the paper.

Disclosure statement

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