I'm trying to figure out the smallest enlargement factor which I need to apply to one ellipsoid $E_1$ in order to fit another one $E_2$. Precisely, let $E(c, S) := \{x | (x-c)^T S (x-c) \leq 1\}$ be an ellipsoid with s.p.d. shape matrix $S$ centered at $c$.
How do I parametrically minimize $\alpha \in \mathbb{R}^+_0$ such that $E_1 := E(0, \alpha \cdot S_1) \supseteq E(c, S_2) =: E_2$? If there is no parametric solution available, do you know of any upper bound for $\alpha$ given $c$, $S_1$, and $S_2$?
As a first approach, I considered the following concave quadratic program:
$$ \max\limits_{x} \; \frac{1}{2}\alpha = \max\limits_{x} \; \frac{1}{2} (x+c)^T S_1 \, (x+c) \quad s.t. \; x^T S_2 \, x \leq 1. $$
From geometric intuition, I assume that the inequality constraint must hold strictly, thus I replaced $\leq$ by $=$ in order to avoid having to deal with the KKT conditions. Further, I've read that a concave program on a convex feasible set has a unique solution, however I'm not sure if this holds in general. For the problem at hand it seems to be adequate that there is single maximum. Are those assumption reasonable or will higher dimensions lead to problems?
Solving this analytically, the stationarity condition reads
$$ \nabla\mathcal{L}(x, \lambda) = -(S_1 + \lambda I) x - S_1 \, c \overset{!}{=} 0 \Leftrightarrow x \overset{!}{=} -(S_1 + \lambda I)^{-1}S_1 \, c. $$
This looks like some sort of perturbed Eigenvalue problem to me, however, I'm not sure how to solve it. Plugging this into the constraint leads requires that
$$ c^T S_1 (S_1 + \lambda I)^{-1} S_2 (S_1 + \lambda I)^{-1} S_1 c \overset{!}{=} 1 $$
which doesn't appear to be readily solvable for $\lambda$ either.
I'm glad for any suggestions. Thanks in advance!