Hi I would like to solve the following optimization problem.

Let $A$ be an $n$ by $n$ nonnegative real matrix where $A^{-1}$is an M-matrix. Let $D=\text{diag}\{d_{1}, \dots , d_{n}\}$ is a nonnegative real diagonal matrix. $\rho(DA)$ is 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: minimise $\rho(DA)$ subject to $d_{i}\geq 0$, $i= 1, \dots n$, $\sum_{i=1}^{n}d_{i} = b$

I thought of solving this problem using the optimal 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:

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

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