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Extended Lorentz Cones and Variational Inequalities on Cylinders

Author

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  • Sándor Zoltán Németh

    (University of Birmingham)

  • Guohan Zhang

    (University of Birmingham)

Abstract

Solutions of a variational inequality problem defined by a closed and convex set and a mapping are found by imposing conditions for the monotone convergence with respect to a cone of the Picard iteration corresponding to the composition of the projection onto the defining closed and convex set and the difference in the identity mapping and the defining mapping. One of these conditions is the isotonicity of the projection onto the defining closed and convex set. If the closed and convex set is a cylinder and the cone is an extented Lorentz cone, then this condition can be dropped because it is automatically satisfied. In this case, a large class of affine mappings and cylinders which satisfy the conditions of monotone convergence above is presented. The obtained results are further specialized for unbounded box-constrained variational inequalities. In a particular case of a cylinder with a base being a cone, the variational inequality is reduced to a generalized mixed complementarity problem which has been already considered in Németh and Zhang (J Global Optim 62(3):443–457, 2015).

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:168:y:2016:i:3:d:10.1007_s10957-015-0833-6
    DOI: 10.1007/s10957-015-0833-6
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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.
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    Cited by:

    1. 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.
    2. 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.
    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.
    4. 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.
    5. 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.

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