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Constraint selection in a build-up interior-point cutting-plane method for solving relaxations of the stable-set problem

Author

Listed:
  • Alexander Engau
  • Miguel Anjos
  • Immanuel Bomze

Abstract

The stable-set problem is an NP-hard problem that arises in numerous areas such as social networking, electrical engineering, environmental forest planning, bioinformatics clustering and prediction, and computational chemistry. While some relaxations provide high-quality bounds, they result in very large and expensive conic optimization problems. We describe and test an integrated interior-point cutting-plane method that efficiently handles the large number of nonnegativity constraints in the popular doubly-nonnegative relaxation. This algorithm identifies relevant inequalities dynamically and selectively adds new constraints in a build-up fashion. We present computational results showing the significant benefits of this approach in comparison to a standard interior-point cutting-plane method. Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Alexander Engau & Miguel Anjos & Immanuel Bomze, 2013. "Constraint selection in a build-up interior-point cutting-plane method for solving relaxations of the stable-set problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 78(1), pages 35-59, August.
    • Handle: RePEc:spr:mathme:v:78:y:2013:i:1:p:35-59
    DOI: 10.1007/s00186-013-0431-z
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    References listed on IDEAS

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    1. Alexander Engau, 2012. "Recent Progress in Interior-Point Methods: Cutting-Plane Algorithms and Warm Starts," International Series in Operations Research & Management Science, in: Miguel F. Anjos & Jean B. Lasserre (ed.), Handbook on Semidefinite, Conic and Polynomial Optimization, chapter 0, pages 471-498, Springer.
    2. Bomze, Immanuel M., 2012. "Copositive optimization – Recent developments and applications," European Journal of Operational Research, Elsevier, vol. 216(3), pages 509-520.
    3. Klerk, Etienne de, 2010. "Exploiting special structure in semidefinite programming: A survey of theory and applications," European Journal of Operational Research, Elsevier, vol. 201(1), pages 1-10, February.
    4. Miguel F. Anjos & Anthony Vannelli, 2008. "Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes," INFORMS Journal on Computing, INFORMS, vol. 20(4), pages 611-617, November.
    5. John A. Kaliski & Yinyu Ye, 1993. "A Short-Cut Potential Reduction Algorithm for Linear Programming," Management Science, INFORMS, vol. 39(6), pages 757-776, June.
    6. Immanuel Bomze & Werner Schachinger & Gabriele Uchida, 2012. "Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization," Journal of Global Optimization, Springer, vol. 52(3), pages 423-445, March.
    7. Francisco Barahona & Andrés Weintraub & Rafael Epstein, 1992. "Habitat Dispersion in Forest Planning and the Stable Set Problem," Operations Research, INFORMS, vol. 40(1-supplem), pages 14-21, February.
    8. Fischer, I. & Gruber, G. & Rendl, F. & Sotirov, R., 2006. "Computational experience with a bundle approach for semidenfinite cutting plane relaxations of max-cut and equipartition," Other publications TiSEM 03dfd8c3-9216-4c75-8921-3, Tilburg University, School of Economics and Management.
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