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Sufficient weighted complementarity problems

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  • Florian A. Potra

    (University of Maryland, Baltimore County)

Abstract

This paper presents some fundamental results about sufficient linear weighted complementarity problems. Such a problem depends on a nonnegative weight vector. If the weight vector is zero, the problem reduces to a sufficient linear complementarity problem that has been extensively studied. The introduction of the more general notion of a weighted complementarity problem (wCP) was motivated by the fact that wCP can model more general equilibrium problems than the classical complementarity problem (CP). The introduction of a nonzero weight vector makes the theory of wCP more complicated than the theory of CP. The paper gives a characterization of sufficient linear wCP and proposes a corrector–predictor interior-point method for its numerical solution. While the proposed algorithm does not depend on the handicap $$\kappa $$ κ of the problem its computational complexity is proportional with $$1+\kappa $$ 1 + κ . If the weight vector is zero and the starting point is relatively well centered, then the computational complexity of our algorithm is the same as the best known computational complexity for solving sufficient linear CP.

Suggested Citation

  • Florian A. Potra, 2016. "Sufficient weighted complementarity problems," Computational Optimization and Applications, Springer, vol. 64(2), pages 467-488, June.
    • Handle: RePEc:spr:coopap:v:64:y:2016:i:2:d:10.1007_s10589-015-9811-z
    DOI: 10.1007/s10589-015-9811-z
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    References listed on IDEAS

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    1. Filiz Gurtuna & Cosmin Petra & Florian Potra & Olena Shevchenko & Adrian Vancea, 2011. "Corrector-predictor methods for sufficient linear complementarity problems," Computational Optimization and Applications, Springer, vol. 48(3), pages 453-485, April.
    2. Tibor Illés & Marianna Nagy & Tamás Terlaky, 2010. "A polynomial path-following interior point algorithm for general linear complementarity problems," Journal of Global Optimization, Springer, vol. 47(3), pages 329-342, July.
    3. Josef Stoer, 2001. "High Order Long-Step Methods for Solving Linear Complementarity Problems," Annals of Operations Research, Springer, vol. 103(1), pages 149-159, March.
    4. Josef Stoer & Martin Wechs & Shinji Mizuno, 1998. "High Order Infeasible-Interior-Point Methods for Solving Sufficient Linear Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 832-862, November.
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    Cited by:

    1. M. Seetharama Gowda, 2019. "Weighted LCPs and interior point systems for copositive linear transformations on Euclidean Jordan algebras," Journal of Global Optimization, Springer, vol. 74(2), pages 285-295, June.
    2. Jingyong Tang & Hongchao Zhang, 2021. "A Nonmonotone Smoothing Newton Algorithm for Weighted Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 189(3), pages 679-715, June.
    3. Soodabeh Asadi & Zsolt Darvay & Goran Lesaja & Nezam Mahdavi-Amiri & Florian Potra, 2020. "A Full-Newton Step Interior-Point Method for Monotone Weighted Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 864-878, September.
    4. Xiaoni Chi & M. Seetharama Gowda & Jiyuan Tao, 2019. "The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra," Journal of Global Optimization, Springer, vol. 73(1), pages 153-169, January.
    5. Xiaoni Chi & Guoqiang Wang & Goran Lesaja, 2024. "Kernel-Based Full-Newton Step Feasible Interior-Point Algorithm for $$P_{*}(\kappa )$$ P ∗ ( κ ) -Weighted Linear Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 108-132, July.
    6. Jingyong Tang & Jinchuan Zhou, 2021. "Quadratic convergence analysis of a nonmonotone Levenberg–Marquardt type method for the weighted nonlinear complementarity problem," Computational Optimization and Applications, Springer, vol. 80(1), pages 213-244, September.
    7. Jingyong Tang & Jinchuan Zhou & Hongchao Zhang, 2023. "An Accelerated Smoothing Newton Method with Cubic Convergence for Weighted Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 196(2), pages 641-665, February.
    8. Behrouz Kheirfam, 2024. "Complexity Analysis of a Full-Newton Step Interior-Point Method for Monotone Weighted Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 133-145, July.
    9. Yongsheng Rao & Jianwei Su & Behrouz Kheirfam, 2024. "A Full-Newton Step Interior-Point Method for Weighted Quadratic Programming Based on the Algebraic Equivalent Transformation," Mathematics, MDPI, vol. 12(7), pages 1-11, April.
    10. Xiaoni Chi & Guoqiang Wang, 2021. "A Full-Newton Step Infeasible Interior-Point Method for the Special Weighted Linear Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 108-129, July.

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