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Finding the set of global minimizers of a piecewise affine function

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  • Majid E. Abbasov

    (St. Petersburg State University
    Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences)

Abstract

Coexhausters are families of convex compact sets that allow one to represent the approximation of the increment of a function at a given point in the form of minmax or maxmin of affine functions. We demonstrate that this representation can be used to define a piecewise affine function and therefore coexhausters are a natural technique for studying the problem of finding a global minimum of piecewise affine functions. All the conditions and methods in the current study were obtained by means of coexhausters theory. Firstly, we apply coexhauster based conditions to state and prove necessary and sufficient conditions for a piecewise affine function to be bounded from below. Secondly, we use coexhausters to construct a simple method which allows one to get the minimum value of the studied function and the corresponding set of all its global minimizers. Illustrative numerical examples are provided throughout the paper.

Suggested Citation

  • Majid E. Abbasov, 2023. "Finding the set of global minimizers of a piecewise affine function," Journal of Global Optimization, Springer, vol. 85(1), pages 1-13, January.
    • Handle: RePEc:spr:jglopt:v:85:y:2023:i:1:d:10.1007_s10898-022-01191-7
    DOI: 10.1007/s10898-022-01191-7
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    References listed on IDEAS

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    1. M. E. Abbasov & V. F. Demyanov, 2013. "Adjoint Coexhausters in Nonsmooth Analysis and Extremality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 535-553, March.
    2. 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.
    3. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico & Adil M. Bagirov, 2018. "Minimizing nonsmooth DC functions via successive DC piecewise-affine approximations," Journal of Global Optimization, Springer, vol. 71(1), pages 37-55, May.
    4. Stephan Dempe & Felix Harder & Patrick Mehlitz & Gerd Wachsmuth, 2019. "Solving inverse optimal control problems via value functions to global optimality," Journal of Global Optimization, Springer, vol. 74(2), pages 297-325, June.
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