SDPs with complex numbers

For mysterious reasons, some time ago I found myself reading SeDuMi’s manual. To my surprise, it claimed to support SDPs with complex numbers. More specifically, it could handle positive semidefiniteness constraints on complex Hermitian matrices, instead of only real symmetric matrices as all other solvers.

I was very excited, because this promised a massive increase in performance for such problems, and in my latest paper I’m solving a massive SDP with complex Hermitian matrices.

The usual way to handle complex problems is to map them into real ones via the transformation
\[ f(M) = \begin{pmatrix} \Re(M) & \Im(M) \\ \Im(M)^T & \Re(M) \end{pmatrix}. \]The spectrum of the $f(M)$ consists of two copies of the spectrum of $M$, and $f(MN) = f(M)f(N)$, so you can see that one can do an exact mapping. The problem is that the matrix is now twice as big: the number of parameters it needs is roughly twice what was needed for the original complex matrix1, so this wastes a bit of memory. More problematic, the interior-point algorithm needs to calculate the Cholesky decomposition, which has complexity $O(d^3)$, so we are slowing the algorithm down by a factor of 8!

I wrote then a trivial SDP to test SeDuMi, and of course it failed. A more careful reading of the documentation showed that I was formatting the input incorrectly, so I fixed that, and it failed again. Reading the documentation again and again convinced me that the input was now correct: it must have been a bug in SeDuMi itself.

Lured by the promise of a 8 times speedup, I decided to dare the dragon, and looked into the source code of SeDuMi. It was written more than 20 years ago, and the original developer is dead, so you might understand why I was afraid. Luckily the code had comments, otherwise how could I figure out what it was supposed to do when it wasn’t doing it?

It turned out to be a simple fix, the real challenge was only understanding what was going on. And the original developer wasn’t to blame, the bug had been introduced by another person in 2017.

Now with SeDuMi working, I proceeded to benchmarking. To my despair, the promised land wasn’t there: there was no difference at all in speed between the complex version and the real version. I was at the point of giving up, when Johan Löfberg, the developer of YALMIP kindly pointed out to me that SeDuMi also needs to do a Cholesky decomposition of the Hessian, a $m \times m$ matrix where $m$ is the number of constraints. The complexity of Sedumi is then roughly $O(m^3 + d^3)$ using complex numbers, and $O(m^3 + 8d^3)$ when solving the equivalent real version. In my test problem I had $m=d^2$ constraints, so no wonder I couldn’t see any speedup.

I wrote then another test SDP, this time with a single constraint, and voilà! There was a speedup of roughly 4 times! Not 8, probably because computing the Cholesky decomposition of a complex matrix is harder than of a real matrix, and there is plenty of other stuff going on, but no matter, a 4 times speedup is nothing to sneer at.

The problem now that this was only when calling SeDuMi directly, which requires writing the SDP in canonical form. I wasn’t going to do that for any nontrivial problem. It’s not hard per se, but requires the patience of a monk. This is why we have preprocessors like YALMIP.

To take advantage of the speedup, I had to adapt YALMIP to handle complex problems. Löfberg is very much alive, which makes things much easier.

As it turned out, YALMIP already supported complex numbers but had it disabled, presumably because of the bug in SeDuMi. What was missing was support for dualization of complex problems, which is important because sometimes the dualized version is much more efficient than the primal one. I went to work on that.

Today Löberg accepted the pull request, so right now you can enjoy the speedup if you use the latest git of SeDuMi and YALMIP. If that’s useful to you please test and report any bugs.

What about my original problem? I benchmarked it, and using the complex version of SeDuMi did give me a speedup of roughly 30%. Not so impressive, but definitely welcome. The problem is that SeDuMi is rather slow, and even using the real mapping MOSEK can solve my problem faster than it.

I don’t think it was pointless going through all that, though. First because there are plenty of people that use SeDuMi, as it’s open source, unlike MOSEK. Second because now the groundwork is laid down, and if another solver appears that can handle complex problems, we will be able to use that capability just by flipping a switch.

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3 Responses to SDPs with complex numbers

  1. As usual, this is an excellent post. Thank you for cataloging your saga! Indeed, as someone who works often with SDPs in quantum information the lack of general support for complex numbers in solvers is a pain point.

    Thank you kindly for your contribution to the community and for your very nice post on your journey–it was a great read.

  2. Jie Wang says:

    Hi, this note might be helpful for solving your complex SDPs:

  3. Mateus Araújo says:

    It’s a nice reformulation, but the main problem is not the extra affine constraints, but the doubling of the dimension of the PSD variable, and this you cannot solve.

    I think the best solution still is to handle complex numbers directly. Note that SeDuMi is not the only solver that can handle it, Hypatia can also do it and it’s a much better solver.

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