Let's say I want to minimize a quadratic form $\sum_{j=1}^n c_jx_j^2$ (all $c_j$ are positive constants), which corresponds to an $n$ dimensional ellipsoid, 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? By analytical I mean "not numerical solutions". Thanks in advance.

There are, you should look at 'Semidefinite Programming': http://en.wikipedia.org/wiki/Semidefinite_programming EDIT: See Pag 58 of L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Rev., vol. 38, pp. 49–95, 1996. regarding results about the problem. 

