Can cvx or tfocs solve convex optimalization with target function complex?


#1

Hello everyone,

I’m trying to solve a basic problem with complex matrix through cvx, the model can be described as follows:
min u’ * M * u
s.t. |ui| = 1, for i = 1~N
and what i write in matlab is (try when N=2):
M = [0.1146 + 0.0191i -0.0591 - 0.0328i
0.0526 + 0.0788i 0.0181 - 0.0813i];
cvx_begin sdp quiet
variable u(2,1);
minimize (real(u’Mu));
subject to
norms(u) = 2;
cvx_end

and the error i get is:
Error using * (line 258)
Disciplined convex programming error:
Invalid quadratic form: product is complex.

Error in testcvx (line 5)
minimize (real(u’Mu));

Is that because of the refuse of complex target function in cvx? If not, could anyone help to show how to correct it?

Thanks in advance.


(Mark L. Stone) #2

M is not Hermitian semidefinite. In fact, it’s not even Hermitian.

Moreover, norm(u) == 1 (or 2) is a non-convex constraint.

norms(u) = 2 is not what you intended. You intended norms(u) == 2, which will be rejected due to being non-convex constraint. norm(u) = 2.would be rejected as an illegal construct.

Using CVX Beta 3.0 Build 1177,
minimize(real(u'*(M+M')*u))
and
minimize(u'*(M+M')*u)
were both accepted by CVX, but I don’t know whey they were because M + M’ has eigenvalues -0.0150, 0.2804; hence is indefinite.
However, these are rejected if u is declared complex (which you did not do).

I am interested in what @mcg has to say about why minimize(u’*(M+M’)*u) was accepted. But if u is declared complex, it is rejected.


#3

oh, Mark, thanks for your detail explanations, i got it:grinning: