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The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra

Author

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  • Xiaoni Chi

    (Guilin University of Electronic Technology)

  • M. Seetharama Gowda

    (University of Maryland, Baltimore County)

  • Jiyuan Tao

    (Loyola University Maryland)

Abstract

A weighted complementarity problem is to find a pair of vectors belonging to the intersection of a manifold and a cone such that the product of the vectors in a certain algebra equals a given weight vector. If the weight vector is zero, we get a complementarity problem. Examples of such problems include the Fisher market equilibrium problem and the linear programming and weighted centering problem. In this paper we consider the weighted horizontal linear complementarity problem in the setting of Euclidean Jordan algebras and establish some existence and uniqueness results. For a pair of linear transformations on a Euclidean Jordan algebra, we introduce the concepts of $$\mathbf{R}_0$$ R 0 , $$\mathbf{R}$$ R , and $$\mathbf{P}$$ P properties and discuss the solvability of wHLCPs under nonzero (topological) degree conditions. A uniqueness result is stated in the setting of $${\mathbb {R}}^{n}$$ R n . We show how our results naturally lead to interior point systems.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:73:y:2019:i:1:d:10.1007_s10898-018-0689-z
    DOI: 10.1007/s10898-018-0689-z
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    References listed on IDEAS

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    1. 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.
    2. M. Seetharama Gowda, 1993. "Applications of Degree Theory to Linear Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 18(4), pages 868-879, November.
    3. Florian A. Potra, 2016. "Sufficient weighted complementarity problems," Computational Optimization and Applications, Springer, vol. 64(2), pages 467-488, June.
    4. M. Seetharama Gowda & Jong-Shi Pang, 1992. "On Solution Stability of the Linear Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 77-83, February.
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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. Punit Kumar Yadav & Palpandi Karuppaiah, 2023. "Generalizations of $$R_0$$ R 0 and $$\textbf{SSM}$$ SSM Properties for Extended Horizontal Linear Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 199(1), pages 392-414, October.
    3. 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.
    4. 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.
    5. 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.
    6. Jingyong Tang & Jinchuan Zhou & Zhongfeng Sun, 2023. "A derivative-free line search technique for Broyden-like method with applications to NCP, wLCP and SI," Annals of Operations Research, Springer, vol. 321(1), pages 541-564, February.

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