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Using quadratic convex reformulation to tighten the convex relaxation of a quadratic program with complementarity constraints

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Abstract

Quadratic Convex Reformulation (QCR) is a technique that has been proposed for binary and mixed integer quadratic programs. In this paper, we extend the QCR method to convex quadratic programs with linear complementarity constraints (QPCCs). Due to the complementarity relationship between the nonnegative variables \(y\) and \(w\), a term \(y^{T}Dw\) can be added to the QPCC objective function, where \(D\) is a nonnegative diagonal matrix chosen to maintain the convexity of the objective function and the global resolution of the QPCC. Following the QCR method, the products of linear equality constraints can also be used to perturb the QPCC objective function, with the goal that the new QP relaxation provides a tighter lower bound. By solving a semidefinite program, an equivalent QPCC can be obtained whose QP relaxation is as tight as possible. In addition, we extend the QCR to a general quadratically constrained quadratic program (QCQP), of which the QPCC is a special example. Computational tests on QPCCs are presented.

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Acknowledgments

We are grateful to two anonymous referees for carefully reading the manuscript and for their helpful comments.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

    Lijie Bai & John E. Mitchell

  2. Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Jong-Shi Pang

Authors
  1. Lijie Bai
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  2. John E. Mitchell
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  3. Jong-Shi Pang
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Correspondence to John E. Mitchell.

Additional information

The work of Mitchell was supported by the National Science Foundation under grant DMS-0715446. The work of Mitchell and Bai was supported by the Air Force Office of Sponsored Research under grant FA9550-08-1-0081 and FA9550-11-1-0260. The work of Pang was supported by the USA. National Science Foundation grant CMMI 0969600 and by the Air Force Office of Sponsored Research under grants FA9550-08-1-0061 and FA9550-11-1-0151.

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Bai, L., Mitchell, J.E. & Pang, JS. Using quadratic convex reformulation to tighten the convex relaxation of a quadratic program with complementarity constraints. Optim Lett 8, 811–822 (2014). https://doi.org/10.1007/s11590-013-0647-0

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  • DOI: https://doi.org/10.1007/s11590-013-0647-0

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