Find smallest ellipsoid centred at 0 that encloses ellipsoid centred at x0

Question

Problem Origin:

I encountered the following mathematical problem when solving a control engineering problem. I strongly believe the problem should be relatively simple, but I can't seem to figure it out.

Definitions:

Define an ellipsoidal set as $E(P,x_0,r) = \{x:(x-x_0)^\top P(x-x_0)\leq r\}$, here $x$ and $x_0$ are in $\mathbb{R}^n$;

Problem:

Consider two ellipsoids $E_1=E(P_1,x_1,r_1)$ and $E_2=E(P_2,0,r_2)$, where $P_1$, $P_2$, $r_1$, and $x_1$ are known.

Find the minimum $r_2$ such that $E_1\subseteq E_2$.

More formally this problem can be written as a constrained optimization problem: $$\begin{array}{ll} \underset{r_2}{\text{minimize}} & r_2\\ \text{subject to} & E_1\subseteq E_2\end{array}$$

Answer I am looking for:

I am looking for one of the following:

1. A method to solve optimization problems with these types of subset inclusion constraints.
2. A method to reformulate the optimization problem to one that has simpler constraints, or might even have an analytic solution.

Answer 1

The optimization problem posed (find the minimal $r_2$ such that $E_1\subseteq E_2$) is equivalent to finding the vector $x$, which is in the inerior of both $E_1$ and $E_2$ that maximizes $r_2$.

Therefore the equivalent optimization problem is:

$$\begin{array}{ll} \underset{x, r_2}{\text{maximize}} & r_2\\ \text{subject to} & (x-x_1)^\top P_1 (x-x_1) \leq r_1\\ & x^\top P_2 x \leq r_2\end{array}$$

This is a quadratically constrained optimization problem for which solvers are widely available. For example solver fmincon in Matlab.