Let's say I want to minimize an $n$ dimensional ellipsoid of the form $\sum_{j=1}^n c_jx_j^2$ (all $c_j$ are positive) over the outer part of the intersection of some given ellipsoids, i.e., minimize subject to the constraints $x^TA_jx \geq 1$, $j=1\ldots m$, where $A_j$ are given positive semidefinite matrices (thus making our region an intersection of ellipsoids). Are there any analytical results for these type of problems? Maybe not for general $A_j$, but at least for some? Thanks in advance.