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Weighted LCPs and interior point systems for copositive linear transformations on Euclidean Jordan algebras

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  • M. Seetharama Gowda

    (University of Maryland, Baltimore County)

Abstract

In the setting of a Euclidean Jordan algebra V with symmetric cone $$V_+$$ V + , corresponding to a linear transformation M, a ‘weight vector’ $$w\in V_+$$ w ∈ V + , and a $$q\in V$$ q ∈ V , we consider the weighted linear complementarity problem wLCP(M, w, q) and (when w is in the interior of $$V_+$$ V + ) the interior point system IPS(M, w, q). When M is copositive on $$V_+$$ V + and q satisfies an interiority condition, we show that both the problems have solutions. A simple consequence, stated in the setting of $$\mathbb {R}^{n}$$ R n is that when M is a copositive plus matrix and q is strictly feasible for the linear complementarity problem LCP(M, q), the corresponding interior point system has a solution. This is analogous to a well-known result of Kojima et al. on $$\mathbf{P}_*$$ P ∗ -matrices and may lead to interior point methods for solving copositive LCPs.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:74:y:2019:i:2:d:10.1007_s10898-019-00760-7
    DOI: 10.1007/s10898-019-00760-7
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    References listed on IDEAS

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    1. C. E. Lemke, 1965. "Bimatrix Equilibrium Points and Mathematical Programming," Management Science, INFORMS, vol. 11(7), pages 681-689, May.
    2. 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.
    3. M. Seetharama Gowda, 1993. "Applications of Degree Theory to Linear Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 18(4), pages 868-879, November.
    4. Florian A. Potra, 2016. "Sufficient weighted complementarity problems," Computational Optimization and Applications, Springer, vol. 64(2), pages 467-488, June.
    5. 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. 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.
    2. 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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