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The convex hull heuristic for nonlinear integer programming problems with linear constraints and application to quadratic 0–1 problems

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  • Monique Guignard

    (University of Pennsylvania)

  • Aykut Ahlatcioglu

    (University of Pennsylvania)

Abstract

The convex hull heuristic is a heuristic for mixed-integer programming problems with a nonlinear objective function and linear constraints. It is a matheuristic in two ways: it is based on the mathematical programming algorithm called simplicial decomposition, or SD (von Hohenbalken in Math Program 13:49–68, 1977), and at each iteration, one solves a mixed-integer programming problem with a linear objective function and the original constraints, and a continuous problem with a nonlinear objective function and a single linear constraint. Its purpose is to produce quickly feasible and often near optimal or optimal solutions for convex and nonconvex problems. It is usually multi-start. We have tested it on a number of hard quadratic 0–1 optimization problems and present numerical results for generalized quadratic assignment problems, cross-dock door assignment problems, quadratic assignment problems and quadratic knapsack problems. We compare solution quality and solution times with results from the literature, when possible.

Suggested Citation

  • Monique Guignard & Aykut Ahlatcioglu, 2021. "The convex hull heuristic for nonlinear integer programming problems with linear constraints and application to quadratic 0–1 problems," Journal of Heuristics, Springer, vol. 27(1), pages 251-265, April.
    • Handle: RePEc:spr:joheur:v:27:y:2021:i:1:d:10.1007_s10732-019-09433-w
    DOI: 10.1007/s10732-019-09433-w
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    References listed on IDEAS

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    1. Monique Guignard, 2003. "Lagrangean relaxation," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 11(2), pages 151-200, December.
    2. Aykut Ahlatçıoğlu & Michael Bussieck & Mustafa Esen & Monique Guignard & Jan-Hendrick Jagla & Alexander Meeraus, 2012. "Combining QCR and CHR for convex quadratic pure 0–1 programming problems with linear constraints," Annals of Operations Research, Springer, vol. 199(1), pages 33-49, October.
    3. Alberto Caprara & David Pisinger & Paolo Toth, 1999. "Exact Solution of the Quadratic Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 11(2), pages 125-137, May.
    4. Jean-François Cordeau & Manlio Gaudioso & Gilbert Laporte & Luigi Moccia, 2006. "A Memetic Heuristic for the Generalized Quadratic Assignment Problem," INFORMS Journal on Computing, INFORMS, vol. 18(4), pages 433-443, November.
    5. Marguerite Frank & Philip Wolfe, 1956. "An algorithm for quadratic programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 95-110, March.
    6. Mauricio G. C. Resende & K. G. Ramakrishnan & Zvi Drezner, 1995. "Computing Lower Bounds for the Quadratic Assignment Problem with an Interior Point Algorithm for Linear Programming," Operations Research, INFORMS, vol. 43(5), pages 781-791, October.
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