Hi I would like to solve the following optimization problem.

Let $A$ be an $n \times n$ nonnegative real matrix where $A^{-1}$is an M-matrix. Let $D=\text{diag}\{d_{1}, \dots , d_{n}\}$ be a nonnegative real diagonal matrix. Let $\rho(DA)$ denote the spectral radius of $DA$.

It is known that $\rho(DA)$ is a convex function over the space of nonnegative diagonal matrices (Friedland(1981), Convex spectral functions. Linear and multilinear algebra, 9,299--316.).

Problem: \begin{equation*} \begin{split} \text{minimise} &\rho(DA)\\ \text{subject to}\quad & d_{i}\geq 0,\ i=1,\ldots n,\quad \sum_{i=1}^{n}d_{i} = b \end{split} \end{equation*}

I thought of solving this problem using the optimality criteria given on page 139 of Boyd and Vandenberghe (2004). The feasible set for this problem is $X=\{D=\text{diag}\{d_{1}, \dots, d_{n}\}| d_{1}\geq 0, \sum_{i=1}^{n}d_{i} = b\}$.

And find a $D^{*} \in X$ such that:

$\langle\nabla\rho(D_{1}A),(D_{1}-D^{*})\rangle\geq 0$

for all $D_{1}\in X$.

My problem is that I do not know how to compute $\nabla\rho(D_{1}A)$.

Question: Is this the right way to approach this problem? If not, could I please be directed to a correct approach.

Thank you