Full Length Paper Series B

Mathematical Programming

, Volume 136, Issue 2, pp 233-251

First online:

On convex relaxations for quadratically constrained quadratic programming

  • Kurt M. AnstreicherAffiliated withDepartment of Management Sciences, University of Iowa Email author 

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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.


Quadratically constrained quadratic programming Convex envelope Semidefinite programming Reformulation-linearization technique

Mathematics Subject Classification

90C26 90C22