I have a problem at the intersection of a range of topics: exponential programming, semi-definite programming and computer science, that I am having trouble finding a decent method for solving.
Take $A_i\in\mathbb{R}^{d\times d}$ with $A_i = A^T_i$, $i\in\mathcal{I}$ and $\mathcal{I}$ is a finite set of indices. $-\infty < \text{tr}(A_i)< 0,\ \forall i\in\mathcal{I}$. We also have $b_i\in\mathbb{R}^+$.
We seek $X_i\in\mathbb{R}^{d \times d}$ that solves the optimization problem
\begin{align} &\min \sum_i b_i e^{\text{tr}(A_i X_i)} \\ \text{s.t.} & \sum_i e^{\text{tr}(X_i)} \leq \mathcal{C} \\ & \| X_i \|^2_{Fr} \leq \alpha\\ & X_i = X^T_i \end{align}
This can be solved using CVX, and thus satisfies Disciplined Convex Programming. The problem is that because it is semi-definite programming on the exponential cone, it requires an approximation of the cone, thus is quite slow. I have also used the large blunt object that is NLOPT which performs quickly, but this seems unsatisfactory given it doesn't really exploit the convex structure.
My question is: What other methods could I reasonably attack this problem with? In particular ones that might work in parallel (the real set $\mathcal{I}$ is sufficiently large that the problem is distributed across many computers).
There are solvers that can work directly with the exponential cone. Look at ECOS and the forthcoming Mosek version 9. If your problems are of a size where interior point methods are viable, one of those solvers might work for you.
What are the actual sizes of your problem instances?