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.
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Acknowledgements
This work was supported in part by National Natural Science Foundation of China (11771255, 11601458, 11431004, 11801325), Chongqing Natural Science Foundation (cstc2018jcyj-yszxX0009) and Shandong Province Natural Science Foundation (ZR2016AM07). The authors are grateful to the anonymous reviewer and the editor for their helpful comments and suggestions.
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Zhu, X., Zhang, J., Zhou, J. et al. Mathematical Programs with Second-Order Cone Complementarity Constraints: Strong Stationarity and Approximation Method. J Optim Theory Appl 181, 521–540 (2019). https://doi.org/10.1007/s10957-018-01464-w
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DOI: https://doi.org/10.1007/s10957-018-01464-w
Keywords
- Mathematical programs with second-order cone complementarity constraints
- Stationarity conditions
- Jordan product
- Calmness conditions
- Approximation methods