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Properties, Extensions and Application of Piecewise Linearization for Euclidean Norm Optimization in $$\mathbb {R}^2$$ R 2

Author

Listed:
  • Aloïs Duguet

    (Université de Toulouse, CNRS, INP)

  • Christian Artigues

    (Université de Toulouse, CNRS, INP)

  • Laurent Houssin

    (Université de Toulouse)

  • Sandra Ulrich Ngueveu

    (Université de Toulouse, CNRS, INP)

Abstract

This work considers nonconvex mixed integer nonlinear programming where nonlinearity comes from the presence of the two-dimensional euclidean norm in the objective or the constraints. We build from the euclidean norm piecewise linearization proposed by Camino et al. (Comput. Optim. Appl. https://doi.org/10.1007/s10589-019-00083-z , 2019) that allows to solve such nonconvex problems via mixed-integer linear programming with an arbitrary approximation guarantee. Theoretical results are established that prove that this linearization is able to satisfy any given approximation level with the minimum number of pieces. An extension of the piecewise linearization approach is proposed. It shares the same theoretical properties for elliptic constraints and/or objective. An application shows the practical appeal of the elliptic linearization on a nonconvex beam layout mixed optimization problem coming from an industrial application.

Suggested Citation

  • Aloïs Duguet & Christian Artigues & Laurent Houssin & Sandra Ulrich Ngueveu, 2022. "Properties, Extensions and Application of Piecewise Linearization for Euclidean Norm Optimization in $$\mathbb {R}^2$$ R 2," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 418-448, November.
    • Handle: RePEc:spr:joptap:v:195:y:2022:i:2:d:10.1007_s10957-022-02083-2
    DOI: 10.1007/s10957-022-02083-2
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    References listed on IDEAS

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    1. Josef Kallrath & Steffen Rebennack, 2014. "Cutting ellipses from area-minimizing rectangles," Journal of Global Optimization, Springer, vol. 59(2), pages 405-437, July.
    2. Steffen Rebennack & Vitaliy Krasko, 2020. "Piecewise Linear Function Fitting via Mixed-Integer Linear Programming," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 507-530, April.
    3. Steffen Rebennack & Josef Kallrath, 2015. "Continuous Piecewise Linear Delta-Approximations for Univariate Functions: Computing Minimal Breakpoint Systems," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 617-643, November.
    4. Steffen Rebennack & Josef Kallrath, 2015. "Continuous Piecewise Linear Delta-Approximations for Bivariate and Multivariate Functions," Journal of Optimization Theory and Applications, Springer, vol. 167(1), pages 102-117, October.
    5. Silva, Thiago Lima & Camponogara, Eduardo, 2014. "A computational analysis of multidimensional piecewise-linear models with applications to oil production optimization," European Journal of Operational Research, Elsevier, vol. 232(3), pages 630-642.
    6. Ngueveu, Sandra Ulrich, 2019. "Piecewise linear bounding of univariate nonlinear functions and resulting mixed integer linear programming-based solution methods," European Journal of Operational Research, Elsevier, vol. 275(3), pages 1058-1071.
    7. Jean-Thomas Camino & Christian Artigues & Laurent Houssin & Stéphane Mourgues, 2019. "Linearization of Euclidean norm dependent inequalities applied to multibeam satellites design," Computational Optimization and Applications, Springer, vol. 73(2), pages 679-705, June.
    8. Rovatti, Riccardo & D’Ambrosio, Claudia & Lodi, Andrea & Martello, Silvano, 2014. "Optimistic MILP modeling of non-linear optimization problems," European Journal of Operational Research, Elsevier, vol. 239(1), pages 32-45.
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    Cited by:

    1. Ploussard, Quentin, 2024. "Piecewise linear approximation with minimum number of linear segments and minimum error: A fast approach to tighten and warm start the hierarchical mixed integer formulation," European Journal of Operational Research, Elsevier, vol. 315(1), pages 50-62.
    2. Wu, Yaqing & Maravelias, Christos T., 2024. "Piecewise linear trees as surrogate models for system design and planning under high-frequency temporal variability," European Journal of Operational Research, Elsevier, vol. 315(2), pages 541-552.

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