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Isotonicity of the Metric Projection and Complementarity Problems in Hilbert Spaces

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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.

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Acknowledgements

The authors would like to thank the referee for his/her very important comments that improved the results and the quality of the paper. The authors were supported financially by the National Natural Science Foundation of China (11371221, 11571296), the Natural Science Foundation of Shandong Province of China (ZR201702170311) and the China Postdoctoral Science Foundation (2017M612307).

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Authors and Affiliations

  1. College of Information Science and Engineering, Shandong Agricultural University, Taian, Shandong, China

    Dezhou Kong

  2. School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, China

    Dezhou Kong & Lishan Liu

  3. Department of Mathematics and Statistics, Curtin University, Perth, Australia

    Lishan Liu & Yonghong Wu

Authors
  1. Dezhou Kong
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  2. Lishan Liu
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  3. Yonghong Wu
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Correspondence to Lishan Liu.

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Kong, D., Liu, L. & Wu, Y. Isotonicity of the Metric Projection and Complementarity Problems in Hilbert Spaces. J Optim Theory Appl 175, 341–355 (2017). https://doi.org/10.1007/s10957-017-1162-8

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  • DOI: https://doi.org/10.1007/s10957-017-1162-8

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