Weird behavior when minimizing norm_nuc

1

I apologize in advance if I’m doing something extremely silly here, but I can’t figure out what is causing some weird behaviour on one of my CVX problems. Here is a minimal CVX example that displays the weirdness that I’m encountering:

cvx_begin sdp quiet
  variable X(2,2) hermitian
  minimize norm_nuc(X)
  subject to
    trace(X) == 1;
    X >= 0;
    X(1,2) == 0.2*1i;
cvx_end

The above code should return a value of 1, since the nuclear norm of every positive semidefinite matrix is just equal to its trace, which is constrained to be 1 here. However, when I run this code (on MATLAB R2013a, using CVX 2.1), I get a value of 1.4000 in cvx_optval. Nonetheless, subsequently running

norm_nuc(X)

on the output of the above CVX problem gives a value of 1.000, as expected. Where is the 1.4000 coming from?

The problem seems to be with the 0.2*1i constraint. If I change that to a real number, then this problem goes away. Also, I get the same weirdness whether I use SDPT3 or SeDuMi as the solver.

2

Sorry to bump this topic, but is there any update or something obvious that I’m missing for this? Over 2 years later, this is still causing me major problems. Minimizing pretty much anything involving norm_nuc or lambda_sum_largest involving complex Hermitian matrices gives incorrect results, regardless of which solver I use. But putting the same model into YALMIP produces correct results.

3

I just ran it using sedumi and sdpt3 under CVX 2.1 build 1116 under MATLAB R2013A; and using sedumi, sdpt3, and scs under CVX 3.0beta build 1177 under MATLAB R2014A. In all cases I got cvx_optval 1.

However, X(1,2) == 0.2*1i was not satisfied in the CVX 3.0beta runs. It was satisfied in the CVX2.1 runs. CVX 3.0beta is know to have bugs, so here is a manifestation of one (or more) of them.

4

Thanks Mark. I’m running CVX 2.1 build 1110, but under a much older version of MATLAB (r2008b). I’ll assume for now that the old version of MATLAB is the culprit, and try to get my hands on a newer one.

5

I cannot promise that the older version of MATLAB is the culprit, but what I can tell you is that I can’t afford to verify correct operation on MATLAB versions that old. I try to go 5 years back but even that is difficult.