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Vector-Valued Implicit Lagrangian For Symmetric Cone Complementarity Problems

Author

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  • LINGCHEN KONG

    (Department of Applied Mathematics, Beijing Jiaotong University, Beijing 100044, P. R. China;
    Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada)

  • LEVENT TUNÇEL

    (Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada)

  • NAIHUA XIU

    (Department of Applied Mathematics, Beijing Jiaotong University, Beijing 100044, P. R. China)

Abstract

The implicit Lagrangian was first proposed by Mangasarian and Solodov as a smooth merit function for the nonnegative orthant complementarity problem. It has attracted much attention in the past ten years because of its utility in reformulating complementarity problems as unconstrained minimization problems. In this paper, exploiting the Jordan-algebraic structure, we extend it to the vector-valued implicit Lagrangian for symmetric cone complementary problem (SCCP), and show that it is a continuously differentiable complementarity function for SCCP and whose Jacobian is strongly semismooth. As an application, we develop the real-valued implicit Lagrangian and the corresponding smooth merit function for SCCP, and give a necessary and sufficient condition for the stationary point of the merit function to be a solution of SCCP. Finally, we show that this merit function can provide a global error bound for SCCP with the uniform Cartesian P-property.

Suggested Citation

  • Lingchen Kong & Levent Tunçel & Naihua Xiu, 2009. "Vector-Valued Implicit Lagrangian For Symmetric Cone Complementarity Problems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 26(02), pages 199-233.
    • Handle: RePEc:wsi:apjorx:v:26:y:2009:i:02:n:s0217595909002171
    DOI: 10.1142/S0217595909002171
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    References listed on IDEAS

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    1. Defeng Sun & Jie Sun, 2008. "Löwner's Operator and Spectral Functions in Euclidean Jordan Algebras," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 421-445, May.
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    Cited by:

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    2. Nan Lu & Zheng-Hai Huang, 2014. "A Smoothing Newton Algorithm for a Class of Non-monotonic Symmetric Cone Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 446-464, May.

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