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Deterministic upper bounds for spatial branch-and-bound methods in global minimization with nonconvex constraints

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Abstract

We discuss some difficulties in determining valid upper bounds in spatial branch-and-bound methods for global minimization in the presence of nonconvex constraints. In fact, two examples illustrate that standard techniques for the construction of upper bounds may fail in this setting. Instead, we propose to perturb infeasible iterates along Mangasarian–Fromovitz directions to feasible points whose objective function values serve as upper bounds. These directions may be calculated by the solution of a single linear optimization problem per iteration. Preliminary numerical results indicate that our enhanced algorithm solves optimization problems where a standard branch-and-bound method does not converge to the correct optimal value.

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Acknowledgments

We would like to thank the two anonymous referees for their precise and substantial remarks which helped to significantly improve the paper.

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Authors and Affiliations

  1. Institute of Operations Research, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

    Peter Kirst & Oliver Stein

  2. Blumenstraße 5, 41363, Jüchen, Germany

    Paul Steuermann

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  1. Peter Kirst
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  2. Oliver Stein
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  3. Paul Steuermann
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Correspondence to Oliver Stein.

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Kirst, P., Stein, O. & Steuermann, P. Deterministic upper bounds for spatial branch-and-bound methods in global minimization with nonconvex constraints. TOP 23, 591–616 (2015). https://doi.org/10.1007/s11750-015-0387-7

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  • DOI: https://doi.org/10.1007/s11750-015-0387-7

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