# Closed form solution to projection on interior of ellipsoid

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!

• An old question of mine is possibly related: Maximizing a quadratic subject to $\|x\|_2\le1$ – user856 Dec 23 '15 at 1:00
• Hmm, I think you're's right. I also was secretly hoping for a clean solution but it may be unavoidably messy. thanks for the pointer! – Y. S. Dec 23 '15 at 1:06