On convex relaxations for quadratically constrained quadratic programming

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.