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