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Tightening a copositive relaxation for standard quadratic optimization problems

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Abstract

We focus in this paper the problem of improving the semidefinite programming (SDP) relaxations for the standard quadratic optimization problem (standard QP in short) that concerns with minimizing a quadratic form over a simplex. We first analyze the duality gap between the standard QP and one of its SDP relaxations known as “strengthened Shor’s relaxation”. To estimate the duality gap, we utilize the duality information of the SDP relaxation to construct a graph G ∗. The estimation can be then reduced to a two-phase problem of enumerating first all the minimal vertex covers of G ∗ and solving next a family of second-order cone programming problems. When there is a nonzero duality gap, this duality gap estimation can lead to a strictly tighter lower bound than the strengthened Shor’s SDP bound. With the duality gap estimation improving scheme, we develop further a heuristic algorithm for obtaining a good approximate solution for standard QP.

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Acknowledgements

The authors are grateful to the two anonymous referees for their valuable comments and suggestions that have greatly helped the authors improve the paper.

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Authors and Affiliations

  1. State Key Laboratory of Software Development Environment, LMIB of the Ministry of Education, School of Mathematics and System Sciences, Beihang University, Beijing, 100191, P.R. China

    Yong Xia

  2. Department of Mathematics, National Cheng Kung University, Tainan, Taiwan

    Ruey-Lin Sheu

  3. Department of Management Science, School of Management, Fudan University, Shanghai, 200433, P.R. China

    Xiaoling Sun

  4. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    Duan Li

Authors
  1. Yong Xia
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  2. Ruey-Lin Sheu
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  3. Xiaoling Sun
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  4. Duan Li
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Corresponding author

Correspondence to Ruey-Lin Sheu.

Additional information

This research was supported by Taiwan NSC 98-2115-M-006-010-MY2, by National Center for Theoretical Sciences of Taiwan(South), by Research Grants Council of Hong Kong under grants 414207, 414808 and 414610, by National Natural Science Foundation of China under grants 10971034, 70832002 and 11001006, by the fund of State Key Laboratory of Software Development Environment under grant SKLSDE-2011ZX-15.

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Xia, Y., Sheu, RL., Sun, X. et al. Tightening a copositive relaxation for standard quadratic optimization problems. Comput Optim Appl 55, 379–398 (2013). https://doi.org/10.1007/s10589-012-9522-7

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