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A method for convex black-box integer global optimization

Author

Listed:
  • Jeffrey Larson

    (Argonne National Laboratory)

  • Sven Leyffer

    (Argonne National Laboratory)

  • Prashant Palkar

    (Indian Institute of Technology Bombay)

  • Stefan M. Wild

    (Argonne National Laboratory)

Abstract

We study the problem of minimizing a convex function on a nonempty, finite subset of the integer lattice when the function cannot be evaluated at noninteger points. We propose a new underestimator that does not require access to (sub)gradients of the objective; such information is unavailable when the objective is a blackbox function. Rather, our underestimator uses secant linear functions that interpolate the objective function at previously evaluated points. These linear mappings are shown to underestimate the objective in disconnected portions of the domain. Therefore, the union of these conditional cuts provides a nonconvex underestimator of the objective. We propose an algorithm that alternates between updating the underestimator and evaluating the objective function. We prove that the algorithm converges to a global minimum of the objective function on the feasible set. We present two approaches for representing the underestimator and compare their computational effectiveness. We also compare implementations of our algorithm with existing methods for minimizing functions on a subset of the integer lattice. We discuss the difficulty of this problem class and provide insights into why a computational proof of optimality is challenging even for moderate problem sizes.

Suggested Citation

  • Jeffrey Larson & Sven Leyffer & Prashant Palkar & Stefan M. Wild, 2021. "A method for convex black-box integer global optimization," Journal of Global Optimization, Springer, vol. 80(2), pages 439-477, June.
    • Handle: RePEc:spr:jglopt:v:80:y:2021:i:2:d:10.1007_s10898-020-00978-w
    DOI: 10.1007/s10898-020-00978-w
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    References listed on IDEAS

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    1. Peter A. Graf & Stephen Billups, 2017. "MDTri: robust and efficient global mixed integer search of spaces of multiple ternary alloys," Computational Optimization and Applications, Springer, vol. 68(3), pages 671-687, December.
    2. Giampaolo Liuzzi & Stefano Lucidi & Francesco Rinaldi, 2015. "Derivative-Free Methods for Mixed-Integer Constrained Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 933-965, March.
    3. Christoph Buchheim & Renke Kuhlmann & Christian Meyer, 2018. "Combinatorial optimal control of semilinear elliptic PDEs," Computational Optimization and Applications, Springer, vol. 70(3), pages 641-675, July.
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    Cited by:

    1. Tommaso Giovannelli & Giampaolo Liuzzi & Stefano Lucidi & Francesco Rinaldi, 2022. "Derivative-free methods for mixed-integer nonsmooth constrained optimization," Computational Optimization and Applications, Springer, vol. 82(2), pages 293-327, June.
    2. ten Eikelder, S.C.M. & van Amerongen, J.H.M., 2023. "Resource allocation problems with expensive function evaluations," European Journal of Operational Research, Elsevier, vol. 306(3), pages 1170-1185.

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