calculate diamond norm using dual problem

[benkj]

This implements the same algorithm of QETLAB, first discussed in this paper, section 3

Fix #53

[pgawron]

Test for diamond distance symmetry should be added to the test/convex.jl

[lpawela]

Shouldn't it be simply Z in :SDP instead of Z+Z' in :SDP? Except fixing a bug, which I cannot reproduce, does the dual problem have advantages over the primal one? Better numerical performance?

[benkj]

Shouldn't it be simply Z in :SDP instead of Z+Z' in :SDP? Except fixing a bug, which I cannot reproduce, does the dual problem have advantages over the primal one? Better numerical performance?

I took inspiration from qetlab where they have the similar line

       cons = [Y0,-Phi;-Phi',Y1];
        cons + cons' >= 0; % avoid some numerical problems: CVX often thinks things aren't symmetric without this

I also did notice some instability with just Z in :SDP.

I was just playing with primal & dual formulation, and found that the dual one didn't have the buggy behaviour of #53 . Not sure if it has other advantages.

[lpawela]

I think we should keep both versions, norm_diamond should take an optional parameter likedual=false setting which approach is used.

[benkj]

I agree that we should keep both versions. We should also force the correct eps. Why did you choose eps=1e-7?

[lpawela]

After some tests on this problem anything higher causes to hit max_iters limit. Also, I recall running into problems in other optimizations when eps was set to something smaller.

[benkj]

Ok. Does it make sense then do add eps as a parameter?
In case we can hit the max_iter with larger dimensional problems.

[lpawela]

Yes, the user should be able to set eps and max_iters.

[benkj]

In the new commit I added a test from #53 and implemented both the primal and dual problem with optional argument

[pgawron]

I think we should keep both versions, norm_diamond should take an optional parameter likedual=false setting which approach is used.

We should have method=:primal as default with option for method=:dual. I believe it would be more Julianesque and elegant.

[benkj]

I have implemented the 'julianeque' notation with method=:primal as default choice