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

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  • Peter Kirst
  • Oliver Stein
  • Paul Steuermann

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. Copyright Sociedad de Estadística e Investigación Operativa 2015

Suggested Citation

  • Peter Kirst & Oliver Stein & Paul Steuermann, 2015. "Deterministic upper bounds for spatial branch-and-bound methods in global minimization with nonconvex constraints," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 591-616, July.
    • Handle: RePEc:spr:topjnl:v:23:y:2015:i:2:p:591-616
    DOI: 10.1007/s11750-015-0387-7
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    References listed on IDEAS

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    1. Sonia Cafieri & Jon Lee & Leo Liberti, 2010. "On convex relaxations of quadrilinear terms," Journal of Global Optimization, Springer, vol. 47(4), pages 661-685, August.
    2. R. Misener & C. A. Floudas, 2010. "Piecewise-Linear Approximations of Multidimensional Functions," Journal of Optimization Theory and Applications, Springer, vol. 145(1), pages 120-147, April.
    3. Ruth Misener & Christodoulos Floudas, 2013. "GloMIQO: Global mixed-integer quadratic optimizer," Journal of Global Optimization, Springer, vol. 57(1), pages 3-50, September.
    4. Ralph Kearfott, 2014. "On rigorous upper bounds to a global optimum," Journal of Global Optimization, Springer, vol. 59(2), pages 459-476, July.
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    Cited by:

    1. Fengqiao Luo & Sanjay Mehrotra, 2021. "A geometric branch and bound method for robust maximization of convex functions," Journal of Global Optimization, Springer, vol. 81(4), pages 835-859, December.
    2. Gabriele Eichfelder & Oliver Stein & Leo Warnow, 2024. "A Solver for Multiobjective Mixed-Integer Convex and Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1736-1766, November.
    3. Peter Kirst & Fabian Rigterink & Oliver Stein, 2017. "Global optimization of disjunctive programs," Journal of Global Optimization, Springer, vol. 69(2), pages 283-307, October.
    4. Gabriele Eichfelder & Peter Kirst & Laura Meng & Oliver Stein, 2021. "A general branch-and-bound framework for continuous global multiobjective optimization," Journal of Global Optimization, Springer, vol. 80(1), pages 195-227, May.
    5. Gabriele Eichfelder & Kathrin Klamroth & Julia Niebling, 2021. "Nonconvex constrained optimization by a filtering branch and bound," Journal of Global Optimization, Springer, vol. 80(1), pages 31-61, May.
    6. Marendet, Antoine & Goldsztejn, Alexandre & Chabert, Gilles & Jermann, Christophe, 2020. "A standard branch-and-bound approach for nonlinear semi-infinite problems," European Journal of Operational Research, Elsevier, vol. 282(2), pages 438-452.
    7. Peter Kirst & Oliver Stein, 2019. "Global optimization of generalized semi-infinite programs using disjunctive programming," Journal of Global Optimization, Springer, vol. 73(1), pages 1-25, January.

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