On convex relaxations for quadratically constrained quadratic programming
 Kurt M. Anstreicher
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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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 Title
 On convex relaxations for quadratically constrained quadratic programming
 Journal

Mathematical Programming
Volume 136, Issue 2 , pp 233251
 Cover Date
 20121201
 DOI
 10.1007/s1010701206023
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Quadratically constrained quadratic programming
 Convex envelope
 Semidefinite programming
 Reformulationlinearization technique
 90C26
 90C22
 Industry Sectors
 Authors

 Kurt M. Anstreicher ^{(1)}
 Author Affiliations

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