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Mathematical Programs with Second-Order Cone Complementarity Constraints: Strong Stationarity and Approximation Method

Author

Listed:
  • Xide Zhu

    (Shanghai University
    Yokohama National University)

  • Jin Zhang

    (Southern University of Science and Technology)

  • Jinchuan Zhou

    (Shandong University of Technology)

  • Xinmin Yang

    (Chongqing Normal University)

Abstract

The existence of complementarity constraints causes the difficulties for studying mathematical programs with second-order cone complementarity constraints, since the standard constraint qualification, such as Robinson’s constraint qualification, is invalid. Therefore, various stationary conditions including strong, Mordukhovich and Clarke stationary conditions have been proposed, according to different reformulations of the second-order cone complementarity constraints. In this paper, we present a new reformulation of this problem by taking into consideration the Jordan algebra associated with the second-order cone. It ensures that the classical Karush–Kuhn–Tucker condition coincides with the strong stationary condition of the original problem. Furthermore, we propose a class of approximation methods to solve mathematical programs with second-order cone complementarity constraints. Any accumulation point of the iterative sequences, generated by the approximation method, is Clarke stationary under the corresponding linear independence constraint qualification. This stationarity can be enhanced to strong stationarity with an extra strict complementarity condition. Preliminary numerical experiments indicate that the proposed method is effective.

Suggested Citation

  • Xide Zhu & Jin Zhang & Jinchuan Zhou & Xinmin Yang, 2019. "Mathematical Programs with Second-Order Cone Complementarity Constraints: Strong Stationarity and Approximation Method," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 521-540, May.
    • Handle: RePEc:spr:joptap:v:181:y:2019:i:2:d:10.1007_s10957-018-01464-w
    DOI: 10.1007/s10957-018-01464-w
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    References listed on IDEAS

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    1. Holger Scheel & Stefan Scholtes, 2000. "Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 1-22, February.
    2. Jia Wu & Liwei Zhang & Yi Zhang, 2013. "A smoothing Newton method for mathematical programs governed by second-order cone constrained generalized equations," Journal of Global Optimization, Springer, vol. 55(2), pages 359-385, February.
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