# 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

## 1 Answer

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?

• The optimization arises in a finite element post process, so its typically of the order of number of elements in the mesh. This can be up in the 1e7 territory, at which point the mesh is already distributed over a large number processors. I'm investigating the TAO optimizer for the time being, as it seems to be the parallel optimizer of choice. – NeedsToKnowMoreMaths Feb 27 at 14:26