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Extended Lorentz cones and mixed complementarity problems

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  • S. Németh
  • G. Zhang

Abstract

In this paper we extend the notion of a Lorentz cone in a Euclidean space as follows: we divide the index set corresponding to the coordinates of points in two disjoint classes. By definition a point belongs to an extended Lorentz cone associated with this division, if the coordinates corresponding to one class are at least as large as the norm of the vector formed by the coordinates corresponding to the other class. We call a closed convex set isotone projection set with respect to a pointed closed convex cone if the projection onto the set is isotone (i.e., order preserving) with respect to the partial order defined by the cone. We determine the isotone projection sets with respect to an extended Lorentz cone. In particular, a Cartesian product between an Euclidean space and any closed convex set in another Euclidean space is such a set. We use this property to find solutions of general mixed complementarity problems recursively. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • S. Németh & G. Zhang, 2015. "Extended Lorentz cones and mixed complementarity problems," Journal of Global Optimization, Springer, vol. 62(3), pages 443-457, July.
    • Handle: RePEc:spr:jglopt:v:62:y:2015:i:3:p:443-457
    DOI: 10.1007/s10898-014-0259-y
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Sándor Zoltán Németh & Lianghai Xiao, 2018. "Linear Complementarity Problems on Extended Second Order Cones," Journal of Optimization Theory and Applications, Springer, vol. 176(2), pages 269-288, February.
    2. 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.
    3. Yingchao Gao & Sándor Zoltán Németh & Roman Sznajder, 2022. "The Monotone Extended Second-Order Cone and Mixed Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 381-407, June.
    4. Roman Sznajder, 2016. "The Lyapunov rank of extended second order cones," Journal of Global Optimization, Springer, vol. 66(3), pages 585-593, November.
    5. O. P. Ferreira & S. Z. Németh, 2018. "How to project onto extended second order cones," Journal of Global Optimization, Springer, vol. 70(4), pages 707-718, April.
    6. Sándor Zoltán Németh & Guohan Zhang, 2016. "Extended Lorentz Cones and Variational Inequalities on Cylinders," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 756-768, March.
    7. 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.

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