An algorithm for nonsmooth optimization by successive piecewise linearization

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Abstract

We present an optimization method for Lipschitz continuous, piecewise smooth (PS) objective functions based on successive piecewise linearization. Since, in many realistic cases, nondifferentiabilities are caused by the occurrence of abs(), max(), and min(), we concentrate on these nonsmooth elemental functions. The method’s idea is to locate an optimum of a PS objective function by explicitly handling the kink structure at the level of piecewise linear models. This piecewise linearization can be generated in its abs-normal-form by minor extension of standard algorithmic, or automatic, differentiation tools. In this paper it is shown that the new method when started from within a compact level set generates a sequence of iterates whose cluster points are all Clarke stationary. Numerical results including comparisons with other nonsmooth optimization methods then illustrate the capabilities of the proposed approach.

Keywords

Piecewise smoothness Nonsmooth optimization Algorithmic differentiation Abs-normal form Clarke stationary 

Mathematics Subject Classification

49J52 90C56 
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Notes

Acknowledgements

The authos are indebted to the two referees for their careful reading of the first submission and their many objections and suggestions, which helped to greatly improve the consistency and the readability of the paper.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Department of MathematicsPaderborn UniversityPaderbornGermany
  2. 2.School of Mathematical Science and Information TechnologyYachaytechUrcuquiEcuador

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