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Combining QCR and CHR for convex quadratic pure 0–1 programming problems with linear constraints

Author

Listed:
  • Aykut Ahlatçıoğlu
  • Michael Bussieck
  • Mustafa Esen
  • Monique Guignard
  • Jan-Hendrick Jagla
  • Alexander Meeraus

Abstract

The convex hull relaxation (CHR) method (Albornoz in Doctoral Dissertation, 1998 , Ahlatçıoğlu in Summer paper, 2007 , Ahlatçıoğlu and Guignard in OPIM Dept. Report, 2010 ) provides lower bounds and feasible solutions on convex 0–1 nonlinear programming problems with linear constraints. In the quadratic case, these bounds may often be improved by a preprocessing step that adds to the quadratic objective function terms that are equal to 0 for all 0–1 feasible solutions yet increase its continuous minimum. Prior to computing CHR bounds, one may use Plateau’s quadratic convex reformulation (QCR) method (2006), or one of its weaker predecessors designed for unconstrained problems, the eigenvalue method of Hammer and Rubin (RAIRO 3:67–79, 1970 ) or the method of Billionnet and Elloumi (Math. Program, Ser. A 109:55–68, 2007 ). In this paper, we first describe the CHR method, and then present the QCR reformulation methods. We present computational results for convex GQAP problems. Copyright Springer Science+Business Media, LLC 2012

Suggested Citation

  • 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.
    • Handle: RePEc:spr:annopr:v:199:y:2012:i:1:p:33-49:10.1007/s10479-011-0969-1
    DOI: 10.1007/s10479-011-0969-1
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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. Pessoa, Artur Alves & Hahn, Peter M. & Guignard, Monique & Zhu, Yi-Rong, 2010. "Algorithms for the generalized quadratic assignment problem combining Lagrangean decomposition and the Reformulation-Linearization Technique," European Journal of Operational Research, Elsevier, vol. 206(1), pages 54-63, October.
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    Cited by:

    1. Wu, Baiyi & Li, Duan & Jiang, Rujun, 2019. "Quadratic convex reformulation for quadratic programming with linear on–off constraints," European Journal of Operational Research, Elsevier, vol. 274(3), pages 824-836.
    2. 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.

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