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Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations

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This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of non- convexities: integer variables and non-convex quadratic constraints. To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the lift-and-project methodology. In particular, we propose new methods for generating valid inequalities from the equation Y =  x x T. We use the non-convex constraint \({ Y - x x^T \preccurlyeq 0}\) to derive disjunctions of two types. The first ones are directly derived from the eigenvectors of the matrix Y − x x T with positive eigenvalues, the second type of disjunctions are obtained by combining several eigenvectors in order to minimize the width of the disjunction. We also use the convex SDP constraint \({ Y - x x^T \succcurlyeq 0}\) to derive convex quadratic cuts, and we combine both approaches in a cutting plane algorithm. We present computational results to illustrate our findings.

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Authors and Affiliations

  1. Axioma Inc., 8800 Roswell Road, Building B. Suite 295, Atlanta, GA, 30350, USA

    Anureet Saxena

  2. Laboratoire d’Informatique Fondamentale de Marseille, CNRS-Aix Marseille Universités, Aix-en-Provence, France

    Pierre Bonami

  3. IBM T.J. Watson Research Center, Yorktown Heights, NY, 10598, USA

    Jon Lee

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  1. Anureet Saxena
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  2. Pierre Bonami
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Correspondence to Anureet Saxena.

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For Egon Balas, the father of disjunctive programming.

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Saxena, A., Bonami, P. & Lee, J. Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations. Math. Program. 124, 383–411 (2010). https://doi.org/10.1007/s10107-010-0371-9

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