This may be a silly question, but is there a closed form solution to the projection on the set $\{y | y^TA^{-1}y\leq d\}$? (Here $A$ is symmetric positive definite.)
I got it down to an equivalent optimization problem
minimize $ ||L(u-\hat u)||_2^2$
subject to $||u||_2^2 \leq 1$
where $L$ is a diagonal matrix with only nonnegative values.
(Basically, $A = VDV^T$ the eigenvalue decomposition, $L$ is such that $L^2 = D^{-1}$, and the two change of variables are $u = d^{1/2}L^{-1}V^Ty$, $\hat u = d^{1/2}L^{-1}V^T\hat y$.)
It seems like this last problem should be easy to solve but I can't figure out how. Thoughts? thanks!