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Piecewise-Linear Approximations of Multidimensional Functions

Author

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  • R. Misener

    (Princeton University)

  • C. A. Floudas

    (Princeton University)

Abstract

We develop explicit, piecewise-linear formulations of functions f(x):ℝ n ↦ℝ, n≤3, that are defined on an orthogonal grid of vertex points. If mixed-integer linear optimization problems (MILPs) involving multidimensional piecewise-linear functions can be easily and efficiently solved to global optimality, then non-analytic functions can be used as an objective or constraint function for large optimization problems. Linear interpolation between fixed gridpoints can also be used to approximate generic, nonlinear functions, allowing us to approximately solve problems using mixed-integer linear optimization methods. Toward this end, we develop two different explicit formulations of piecewise-linear functions and discuss the consequences of integrating the formulations into an optimization problem.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:145:y:2010:i:1:d:10.1007_s10957-009-9626-0
    DOI: 10.1007/s10957-009-9626-0
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    References listed on IDEAS

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    1. J. J. H. Forrest & J. P. H. Hirst & J. A. Tomlin, 1974. "Practical Solution of Large Mixed Integer Programming Problems with Umpire," Management Science, INFORMS, vol. 20(5), pages 736-773, January.
    2. Faiz A. Al-Khayyal & James E. Falk, 1983. "Jointly Constrained Biconvex Programming," Mathematics of Operations Research, INFORMS, vol. 8(2), pages 273-286, May.
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    8. Steffen Rebennack & Josef Kallrath, 2012. "Continuous Piecewise Linear δ-Approximations for MINLP Problems. I. Minimal Breakpoint Systems for Univariate Functions," Working Papers 2012-12, Colorado School of Mines, Division of Economics and Business.
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