Comparing different nonsmooth minimization methods and software

Abstract

Most nonsmooth optimization (NSO) methods can be divided into two main groups: subgradient methods and bundle methods. In this paper, we test and compare different methods from both groups as well as some methods which may be considered as hybrids of these two and/or some others. All the solvers tested are so-called general black box methods which, at least in theory, can be applied to solve almost all NSO problems. The test set includes a large number of unconstrained nonsmooth convex and nonconvex problems of different size. In particular, it includes piecewise linear and quadratic problems. The aim of this work is not to foreground some methods over the others but to get some insight on which method to select for certain types of problems.

Keywords:

• nondifferentiable optimization
• bundle methods
• subgradient methods
• numerical performance

Acknowledgements

We are thankful to the anonymous referees for their valuable comments. We would also like to acknowledge Professors A. Kuntsevich and F. Kappel for providing Shor's r-algorithm in their web-page as well as Professors L. Lukšan and J. Vlček for providing the bundle-Newton algorithm. The work was financially supported by the University of Turku (Finland), the University of Ballarat (Australia) and the Australian Research Council.