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Isotonicity of the Metric Projection by Lorentz Cone and Variational Inequalities

Author

Listed:
  • Dezhou Kong

    (Shandong Agricultural University)

  • Lishan Liu

    (Qufu Normal University
    Curtin University)

  • Yonghong Wu

    (Curtin University)

Abstract

In this paper, we first discuss the geometric properties of the Lorentz cone and the extended Lorentz cone. The self-duality and orthogonality of the Lorentz cone are obtained in Hilbert spaces. These properties are fundamental for the isotonicity of the metric projection with respect to the order, induced by the Lorentz cone. According to the Lorentz cone, the quasi-sublattice and the extended Lorentz cone are defined. We also obtain the representation of the metric projection onto cones in Hilbert quasi-lattices. As an application, solutions of the classic variational inequality problem and the complementarity problem are found by the Picard iteration corresponding to the composition of the isotone metric projection onto the defining closed and convex set and the difference in the identity mapping and the defining mapping. Our results generalize and improve various recent results obtained by many others.

Suggested Citation

  • Dezhou Kong & Lishan Liu & Yonghong Wu, 2017. "Isotonicity of the Metric Projection by Lorentz Cone and Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 117-130, April.
    • Handle: RePEc:spr:joptap:v:173:y:2017:i:1:d:10.1007_s10957-017-1084-5
    DOI: 10.1007/s10957-017-1084-5
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    References listed on IDEAS

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    1. S. Németh & G. Zhang, 2015. "Extended Lorentz cones and mixed complementarity problems," Journal of Global Optimization, Springer, vol. 62(3), pages 443-457, July.
    2. Samir Adly & Hadia Rammal, 2015. "A New Method for Solving Second-Order Cone Eigenvalue Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 165(2), pages 563-585, May.
    3. Hiroki Nishimura & Efe A. Ok, 2012. "Solvability of Variational Inequalities on Hilbert Lattices," Mathematics of Operations Research, INFORMS, vol. 37(4), pages 608-625, November.
    4. Pedro Gajardo & Alberto Seeger, 2014. "Equilibrium problems involving the Lorentz cone," Journal of Global Optimization, Springer, vol. 58(2), pages 321-340, February.
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    Cited by:

    1. Dezhou Kong & Lishan Liu & Yonghong Wu, 2017. "Isotonicity of the Metric Projection and Complementarity Problems in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 341-355, November.
    2. Xiao-Peng Yang, 2019. "Evaluation Model and Approximate Solution to Inconsistent Max-Min Fuzzy Relation Inequalities in P2P File Sharing System," Complexity, Hindawi, vol. 2019, pages 1-11, March.
    3. Dezhou Kong & Lishan Liu & Yonghong Wu, 2020. "Isotonicity of Proximity Operators in General Quasi-Lattices and Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 88-104, October.

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