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A modulus-based framework for weighted horizontal linear complementarity problems

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  • Mezzadri, Francesco

Abstract

We develop a modulus-based framework to solve weighted horizontal linear complementarity problems (WHLCPs). First, we reformulate the WHLCP as a modulus-based system whose solution, in general, is not unique. We characterize the solutions by discussing their sign pattern and how they are linked to one another. After this analysis, we exploit the modulus-based formulation to develop new solution methods. In particular, we present a non-smooth Newton iteration and a matrix splitting method for solving WHLCPs. We prove the local convergence of both methods under some assumptions. Finally, we solve numerical experiments involving symmetric and non-symmetric matrices. In this context, we compare our approaches with a recently proposed smoothing Newton's method. The experiments include problems taken from the literature. We also provide numerical insights on relevant parts of the algorithms, such as convergence, attraction basin, and starting iterate.

Suggested Citation

  • Mezzadri, Francesco, 2025. "A modulus-based framework for weighted horizontal linear complementarity problems," Applied Mathematics and Computation, Elsevier, vol. 495(C).
    • Handle: RePEc:eee:apmaco:v:495:y:2025:i:c:s0096300325000402
    DOI: 10.1016/j.amc.2025.129313
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    References listed on IDEAS

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    1. Zheng, Hua & Vong, Seakweng, 2020. "On convergence of the modulus-based matrix splitting iteration method for horizontal linear complementarity problems of H+-matrices," Applied Mathematics and Computation, Elsevier, vol. 369(C).
    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. Masakazu Kojima & Shinji Mizuno & Toshihito Noma, 1990. "Limiting Behavior of Trajectories Generated by a Continuation Method for Monotone Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 662-675, November.
    6. 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.
    7. Zhang, Yongxiong & Zheng, Hua & Lu, Xiaoping & Vong, Seakweng, 2023. "Modulus-based synchronous multisplitting iteration methods without auxiliary variable for solving vertical linear complementarity problems," Applied Mathematics and Computation, Elsevier, vol. 458(C).
    8. Cuiyu Liu & Chenliang Li, 2016. "Synchronous and Asynchronous Multisplitting Iteration Schemes for Solving Mixed Linear Complementarity Problems with H-Matrices," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 169-185, October.
    9. Florian A. Potra, 2016. "Sufficient weighted complementarity problems," Computational Optimization and Applications, Springer, vol. 64(2), pages 467-488, June.
    10. Xia, Zechen & Li, Chenliang, 2015. "Modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problem," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 34-42.
    11. 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.
    12. Zhang, Jian-Jun, 2015. "The relaxed nonlinear PHSS-like iteration method for absolute value equations," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 266-274.
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