Outer approximation of union of ellipsoids via $\log\det$ objective and LMI constraint

Question

Given $A_i$, $b_i$, $c_i$ for $i=1,2,..p$ ($p$ is known), where $A_i\in \mathbb{R^{3\times3}}$ are symmetric positive definite matrices, $b_i\in \mathbb{R^3}$ and $c_i\in \mathbb{R}$. Let the variables of the objective function be $A_0$, $b_0$, and $\tau_1,\tau_2,...\tau_p$, where $A_0\in \mathbb{R^{3\times3}}$ is a symmetric positive definite matrix, $b_0\in \mathbb{R^3}$, and $\tau_1,\tau_2,...\tau_p$ are scalars. The optimization problem can be stated as
$$\min_{A_0, b_0, \tau_1,\tau_2,\dots,\tau_p} {\log\left(\det\left(A_0^{-1}\right)\right)}$$ subject to
$$A_0>0$$ $$\tau_1\geq 0,\tau_2\geq 0,...\tau_p\geq 0$$ $$\begin{bmatrix} A_0 & b_0 & 0\\ b_0^T & -1 & b_0^T\\ 0 & b_0 & -A_0 \end{bmatrix} -\tau_i \begin{bmatrix} A_i & b_i & 0\\ b_i^T & c_i & 0\\ 0 & 0 & 0 \end{bmatrix}\leq 0$$ for $i=1,2, \dots, p$. How to solve this for $A_0, b_0, \tau_1,\tau_2,\dots,\tau_p$?

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Motivation

The application of above optimization problem is in the calculation of minimum volume ellipsoid that contains the union of given ellipsoids, where $A_i, b_i, c_i$ characterizes each given ellipsoids and $A_0, b_0$ characterizes the union ellipsoid.

Answer 1

This is a convex optimization problem. It can be reformulated as maximizing log(det($A_b$)) subject to the nonegativity and LMI constraints. There are built-in functions for log(det) in Disciplined Convex Programming (DCP) convex optimization modeling systems such as CVX (log_det), CVXPY (log_det), and CVXR (log_det), as well as the DCP-capable (and also non-DCP capable) YALMIP (logdet).

Alternatively, it can be reformulated as minimizing det_inv($A_b$)) or maximizing det_rootn($A_b$), where det_inv is the built-in function inverse of determinant and det_rootn is the built-in function nth root of determinant, where n is the number of rows (or columns) of the matrix.