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On convex relaxations for quadratically constrained quadratic programming

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Abstract

We consider convex relaxations for the problem of minimizing a (possibly nonconvex) quadratic objective subject to linear and (possibly nonconvex) quadratic constraints. Let \(\mathcal{F }\) denote the feasible region for the linear constraints. We first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on \(\mathcal{F }\) is dominated by an alternative methodology based on convexifying the range of the quadratic form \(\genfrac(){0.0pt}{}{1}{x}\genfrac(){0.0pt}{}{1}{x}^T\) for \(x\in \mathcal{F }\). We next show that the use of “\(\alpha \)BB” underestimators as computable estimates of convex lower envelopes is dominated by a relaxation of the convex hull of the quadratic form that imposes semidefiniteness and linear constraints on diagonal terms. Finally, we show that the use of a large class of D.C. (“difference of convex”) underestimators is dominated by a relaxation that combines semidefiniteness with RLT constraints.

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Notes

  1. The convex lower envelope is defined at the beginning of the next section.

  2. In the literature, the convex lower envelope of \(f(\cdot )\) is sometimes called simply the convex envelope of \(f(\cdot )\). We prefer to include the word “lower” as a reminder that the convex (lower) envelope is an underestimator of \(f(\cdot )\).

  3. The problem 50-050-1 is structurally similar to the other problems considered. One possibility to attempt to reduce the gap on this problem, which we have not attempted, would be to impose additional valid constraints from the BQP as considered in [21].

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Acknowledgments

I am grateful to two anonymous referees for corrections and suggestions that have improved the paper.

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  1. Department of Management Sciences, University of Iowa, Iowa City, IA, 52242, USA

    Kurt M. Anstreicher

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Anstreicher, K.M. On convex relaxations for quadratically constrained quadratic programming. Math. Program. 136, 233–251 (2012). https://doi.org/10.1007/s10107-012-0602-3

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