= Semidefinite relaxation of slab-constrained quadratic maximization
{title2=$\max\{\operatorname{tr}X:X\succeq0,\ a_i^TXa_i\le1\}$}
Replacing $xx^T$ by a general <positive semidefinite matrix> relaxes maximization of $\|x\|_2^2$ subject to $|a_i^Tx|\le1$. The relaxed value is an upper bound because $xx^T$ has trace $\|x\|_2^2$ and satisfies the same quadratic constraints. Both problems are unbounded if the $a_i$ fail to span the ambient space. If they span it, $H=\sum_i a_ia_i^T$ is positive definite and $\lambda_{\min}(H)\operatorname{tr}X\le\operatorname{tr}(HX)\le m$, proving boundedness and attainment of the relaxation.
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