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Computing tight bounds via piecewise linear functions through the example of circle cutting problems

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  • Steffen Rebennack

    (Colorado School of Mines)

Abstract

This paper discusses approximations of continuous and mixed-integer non-linear optimization problems via piecewise linear functions. Various variants of circle cutting problems are considered, where the non-overlap of circles impose a non-convex feasible region. While the paper is written in an “easy-to-understand” and “hands-on” style which should be accessible to graduate students, also new ideas are presented. Specifically, piecewise linear functions are employed to yield mixed-integer linear programming problems which provide lower and upper bounds on the original problem, the circle cutting problem. The piecewise linear functions are modeled by five different formulations, containing the incremental and logarithmic formulations. Another variant of the cutting problem involves the assignment of circles to pre-defined rectangles. We introduce a new global optimization algorithm, based on piecewise linear function approximations, which converges in finitely many iterations to a globally optimal solution. The discussed formulations are implemented in GAMS. All GAMS-files are available for download in the Electronic supplementary material. Extensive computational results are presented with various illustrations.

Suggested Citation

  • Steffen Rebennack, 2016. "Computing tight bounds via piecewise linear functions through the example of circle cutting problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(1), pages 3-57, August.
    • Handle: RePEc:spr:mathme:v:84:y:2016:i:1:d:10.1007_s00186-016-0546-0
    DOI: 10.1007/s00186-016-0546-0
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    References listed on IDEAS

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    1. Stephen Frank & Steffen Rebennack, 2012. "A Primer on Optimal Power Flow: Theory, Formulation, and Practical Examples," Working Papers 2012-14, Colorado School of Mines, Division of Economics and Business.
    2. Frank, Stephen M. & Rebennack, Steffen, 2015. "Optimal design of mixed AC–DC distribution systems for commercial buildings: A Nonconvex Generalized Benders Decomposition approach," European Journal of Operational Research, Elsevier, vol. 242(3), pages 710-729.
    3. Juan Pablo Vielma & Shabbir Ahmed & George Nemhauser, 2010. "Mixed-Integer Models for Nonseparable Piecewise-Linear Optimization: Unifying Framework and Extensions," Operations Research, INFORMS, vol. 58(2), pages 303-315, April.
    4. Ruth Misener & Christodoulos Floudas, 2014. "ANTIGONE: Algorithms for coNTinuous / Integer Global Optimization of Nonlinear Equations," Journal of Global Optimization, Springer, vol. 59(2), pages 503-526, July.
    5. Ahmet B. Keha & Ismael R. de Farias & George L. Nemhauser, 2006. "A Branch-and-Cut Algorithm Without Binary Variables for Nonconvex Piecewise Linear Optimization," Operations Research, INFORMS, vol. 54(5), pages 847-858, October.
    6. Björn Geißler & Oliver Kolb & Jens Lang & Günter Leugering & Alexander Martin & Antonio Morsi, 2011. "Mixed integer linear models for the optimization of dynamical transport networks," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(3), pages 339-362, June.
    7. Xiang Li & Asgeir Tomasgard & Paul I. Barton, 2011. "Nonconvex Generalized Benders Decomposition for Stochastic Separable Mixed-Integer Nonlinear Programs," Journal of Optimization Theory and Applications, Springer, vol. 151(3), pages 425-454, December.
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    Cited by:

    1. Lambert, Mathieu & Hassani, Rachid, 2023. "Diesel genset optimization in remote microgrids," Applied Energy, Elsevier, vol. 340(C).
    2. John Alasdair Warwicker & Steffen Rebennack, 2022. "A Comparison of Two Mixed-Integer Linear Programs for Piecewise Linear Function Fitting," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 1042-1047, March.
    3. Kazda, Kody & Li, Xiang, 2024. "A linear programming approach to difference-of-convex piecewise linear approximation," European Journal of Operational Research, Elsevier, vol. 312(2), pages 493-511.
    4. Steffen Rebennack, 2022. "Data-driven stochastic optimization for distributional ambiguity with integrated confidence region," Journal of Global Optimization, Springer, vol. 84(2), pages 255-293, October.
    5. Er-Rahmadi, Btissam & Ma, Tiejun, 2022. "Data-driven mixed-Integer linear programming-based optimisation for efficient failure detection in large-scale distributed systems," European Journal of Operational Research, Elsevier, vol. 303(1), pages 337-353.
    6. F. J. Hwang & Yao-Huei Huang, 2021. "An effective logarithmic formulation for piecewise linearization requiring no inequality constraint," Computational Optimization and Applications, Springer, vol. 79(3), pages 601-631, July.
    7. Akang Wang & Chrysanthos E. Gounaris, 2021. "On tackling reverse convex constraints for non-overlapping of unequal circles," Journal of Global Optimization, Springer, vol. 80(2), pages 357-385, June.

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