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On the existence of saddle points for nonlinear second-order cone programming problems

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Abstract

In this paper, we study the existence of local and global saddle points for nonlinear second-order cone programming problems. The existence of local saddle points is developed by using the second-order sufficient conditions, in which a sigma-term is added to reflect the curvature of second-order cone. Furthermore, by dealing with the perturbation of the primal problem, we establish the existence of global saddle points, which can be applicable for the case of multiple optimal solutions. The close relationship between global saddle points and exact penalty representations are discussed as well.

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Acknowledgments

We are gratefully indebted to anonymous referees for their valuable suggestions that help us to essentially improve the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics School of Science, Shandong University of Technology, Zibo, 255049, People’s Republic of China

    Jinchuan Zhou

  2. Department of Mathematics, National Taiwan Normal University, Taipei, 11677, Taiwan

    Jein-Shan Chen

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  1. Jinchuan Zhou
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  2. Jein-Shan Chen
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Correspondence to Jein-Shan Chen.

Additional information

The author’s work is supported by National Natural Science Foundation of China (11101248, 11171247, 11271233), Shandong Province Natural Science Foundation (ZR2010AQ026, ZR2012AM016), and Young Teacher Support Program of Shandong University of Technology. Jein-Shan Chen work is supported by Ministry of Science and Technology, Taiwan.

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Zhou, J., Chen, JS. On the existence of saddle points for nonlinear second-order cone programming problems. J Glob Optim 62, 459–480 (2015). https://doi.org/10.1007/s10898-014-0252-5

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  • DOI: https://doi.org/10.1007/s10898-014-0252-5

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