For a research project I am interested in minimizing $x^TQx$ under the constraints $Qx \le b$, where $Q$ is a positive definite matrix and $b$ is a vector of negative elements. I have solved this with the Gurobi Solver, but as the matrix $Q$ is not sparse, there are lots of terms in the objective. I was wondering if one can take advantage of the constraints being so closely related to the objective.
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Well, setting $y=Q^{\frac{1}{2}}x$ one can minimize $y^Ty$ under the constraints $Q^{\frac{1}{2}}y \le b$. Any better ideas? – Sigurdur Freyr Hafstein Jun 16 '15 at 13:50