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Isotonicity of the Metric Projection and Complementarity Problems in Hilbert Spaces

Author

Listed:
  • Dezhou Kong

    (Shandong Agricultural University
    Qufu Normal University)

  • Lishan Liu

    (Qufu Normal University
    Curtin University)

  • Yonghong Wu

    (Curtin University)

Abstract

In this paper, as the extension of the isotonicity of the metric projection, the isotonicity characterizations with respect to two arbitrary order relations induced by cones of the metric projection operator are studied in Hilbert spaces, when one cone is a subdual cone and some relations between the two orders hold. Moreover, if the metric projection is not isotone in the whole space, we prove that the metric projection is isotone in some domains in both Hilbert lattices and Hilbert quasi-lattices. By using the isotonicity characterizations with respect to two arbitrary order relations of the metric projection, some solvability and approximation theorems for the complementarity problems are obtained. Our results generalize and improve various recent results in the field of study.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:175:y:2017:i:2:d:10.1007_s10957-017-1162-8
    DOI: 10.1007/s10957-017-1162-8
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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. 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.
    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. 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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