I need your expertise in solving the following the follow problem:
Given a convex body P, which is given implicitly by an oracle (either membership or separation), the objective is to find a minimum enclosing ellipsoid using the ellipsoid method.
To achieve such task, we need to formulate the problem such that our variable will represent the ellipsoid:
$$ \min f(x) \\ \text{subject to} \\ x \in \mathbb{R}^{n^2 + n}$$ where x will be something like "$\begin{bmatrix} G \\ c \end{bmatrix}$ " where $G$ is positive definite matrix in $\mathbb{R}^{n \times n}$ and $C$ is a vector which represent a center of an ellipsoid.
What should be given to the ellipsoid method exactly so that it can achieve this purpose?! (what are the constraints? Is $f(x) = \left| G\right|$?)
Please advise.
Thanks in advance.