The Exponential Cone and Semi-definite programming

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).

• Your terminology is a bit off. This isn't semidefinite programming. Semidefinite programs can only have linear matrix inequalities for nonlinearities, and this has none of those. This is simply a smooth nonlinear program. – Michael Grant Jan 12 at 2:52
• And I see nothing whatsoever wrong with using NLOPT, if its performance and accuracy are acceptable. – Michael Grant Jan 12 at 2:54
• Ah thank you, I see now that all the matrix operations can be re-written as linear sums of functions of the indices of the matrices. I'm looking into TAO for the time being. – NeedsToKnowMoreMaths Feb 27 at 14:24