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A simultaneous diagonalization based SOCP relaxation for convex quadratic programs with linear complementarity constraints

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Abstract

This paper proposes a new second-order cone programming (SOCP) relaxation for convex quadratic programs with linear complementarity constraints. The new SOCP relaxation is derived by exploiting the technique that two positive semidefinite matrices can be simultaneously diagonalizable, and is proved to be at least as tight as the classical SOCP relaxation and virtually it can be tighter. We also prove that the proposed SOCP relaxation is equivalent to the semidefinite programming (SDP) relaxation when the objective function is strictly convex. Then an effective branch-and-bound algorithm is designed to find a global optimal solution. Numerical experiments indicate that the proposed SOCP relaxation based branch-and-bound algorithm spends less computing time than the SDP relaxation based branch-and-bound algorithm on condition that the rank of the quadratic objective function is large. The superiority is highlighted when solving the strictly convex quadratic program with linear complementarity constraints.

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Acknowledgements

Zhou’s research has been supported by the National Natural Science Foundation of China under Grant No. 11701512, the Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ16A010010, and the Swiss Government Excellence Scholarship. Xu’s research has been supported by the the National Natural Science Foundation of China under Grant Nos. 11704336, 11647081, and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ15A040003. The authors would like to thank Prof. Martin Jaggi at EPFL for his helpful comments.

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  1. College of Science, Zhejiang University of Technology, Hangzhou, 310023, China

    Jing Zhou & Zhijun Xu

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  1. Jing Zhou
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Correspondence to Zhijun Xu.

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Zhou, J., Xu, Z. A simultaneous diagonalization based SOCP relaxation for convex quadratic programs with linear complementarity constraints. Optim Lett 13, 1615–1630 (2019). https://doi.org/10.1007/s11590-018-1337-8

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