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Second order cone constrained convex relaxations for nonconvex quadratically constrained quadratic programming

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Abstract

In this paper, we present new convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP) problems. While recent research has focused on strengthening convex relaxations of QCQP using the reformulation-linearization technique (RLT), the state-of-the-art methods lose their effectiveness when dealing with (multiple) nonconvex quadratic constraints in QCQP, except for direct lifting and linearization. In this research, we decompose and relax each nonconvex constraint to two second order cone (SOC) constraints and then linearize the products of the SOC constraints and linear constraints to construct some new effective valid constraints. Moreover, we extend the reach of the RLT-like techniques for almost all different types of constraint-pairs (including valid inequalities by linearizing the product of a pair of SOC constraints, and the Hadamard product or the Kronecker product of two respective valid linear matrix inequalities), examine dominance relationships among different valid inequalities, and explore almost all possibilities of gaining benefits from generating valid constraints. We also successfully demonstrate that applying RLT-like techniques to additional redundant linear constraints could reduce the relaxation gap significantly. We demonstrate the efficiency of our results with numerical experiments.

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Acknowledgements

The authors gratefully acknowledge the support of Shanghai Sailing 1004 Program 18YF1401700, Natural Science Foundation of China (NSFC) 11801087 and 11701106, and Hong Kong Research Grants Council under Grant 14213716. The authors would like to express their great appreciation to an anonymous referee for his/her constructive and insightful comments which help improve the paper significantly. The second author is also grateful to the support from Patrick Huen Wing Ming Chair Professorship of Systems Engineering and Engineering Management.

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  1. School of Data Science, Fudan University, Shanghai, China

    Rujun Jiang

  2. School of Data Science, City University of Hong Kong, Hong Kong, China

    Duan Li

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  1. Rujun Jiang
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  2. Duan Li
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Jiang, R., Li, D. Second order cone constrained convex relaxations for nonconvex quadratically constrained quadratic programming. J Glob Optim 75, 461–494 (2019). https://doi.org/10.1007/s10898-019-00793-y

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