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Copositivity tests based on the linear complementarity problem

Author

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  • Carmo Brás
  • Gabriele Eichfelder
  • Joaquim Júdice

Abstract

We present copositivity tests based on new necessary and sufficient conditions which require the solution of linear complementarity problems (LCP). We propose methodologies involving Lemke’s method, an enumerative algorithm and a linear mixed-integer programming formulation to solve the required LCPs. Moreover, we discuss a new necessary condition for (strict) copositivity based on solving a linear program, which can be used as a preprocessing step. The algorithms with these three different variants are thoroughly applied to test matrices from the literature and to max-clique instances with matrices of order up to $$496\times 496$$ 496 × 496 . We compare our procedures with three other copositivity tests from the literature as well as with a general global optimization solver. The numerical results are very promising and equally good and in many cases better than the results reported elsewhere. Copyright Springer Science+Business Media New York 2016

Suggested Citation

  • Carmo Brás & Gabriele Eichfelder & Joaquim Júdice, 2016. "Copositivity tests based on the linear complementarity problem," Computational Optimization and Applications, Springer, vol. 63(2), pages 461-493, March.
    • Handle: RePEc:spr:coopap:v:63:y:2016:i:2:p:461-493
    DOI: 10.1007/s10589-015-9772-2
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    References listed on IDEAS

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    1. Bomze, Immanuel M., 2012. "Copositive optimization – Recent developments and applications," European Journal of Operational Research, Elsevier, vol. 216(3), pages 509-520.
    2. Joaquim Júdice & Ana Faustino & Isabel Ribeiro, 2002. "On the solution of NP-hard linear complementarity problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 10(1), pages 125-145, June.
    3. Julia Sponsel & Stefan Bundfuss & Mirjam Dür, 2012. "An improved algorithm to test copositivity," Journal of Global Optimization, Springer, vol. 52(3), pages 537-551, March.
    4. 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.
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    Cited by:

    1. Orizon Pereira Ferreira & Yingchao Gao & Sándor Zoltán Németh & Petra Renáta Rigó, 2024. "Gradient projection method on the sphere, complementarity problems and copositivity," Journal of Global Optimization, Springer, vol. 90(1), pages 1-25, September.
    2. Janez Povh & Janez Žerovnik, 2021. "On sufficient properties of sufficient matrices," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 29(3), pages 809-822, September.
    3. Darvay, Zsolt & Illés, Tibor & Rigó, Petra Renáta, 2022. "Predictor-corrector interior-point algorithm for P*(κ)-linear complementarity problems based on a new type of algebraic equivalent transformation technique," European Journal of Operational Research, Elsevier, vol. 298(1), pages 25-35.

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