Hello every one,
I am trying to solve a basic SDP problem with complex matrices and get some problem regarding the solution. Here is the problem:
Given two (2x2) hermitian matrices Z1, Z2:
maximise(trace(Z1 E1) + trace(Z2 E2))
with E1 + E2= II (identity), trace(E1)<=1, trace(E2)<= 1, and E1 E2 hermitian semidefinite.
Here is the code:
clc; clear all; close all;
II= eye(2);
Z1 = [0.0206 + 0.0000i 0.0000 + 0.0347i;
0.0000 - 0.0347i -0.0205 + 0.0000i];
Z2 = [-0.0206 + 0.0000i 0.0000 - 0.0347i;
0.0000 + 0.0347i 0.0205 + 0.0000i];
%Solving directly:
[V,D]= eig(Z1);
RHS= abs(D(1,1)-D(2,2));
if (D(1,1)>D(2,2))
P1= V(:,1)*V(:,1)’;
else
P1= V(:,2)*V(:,2)’;
P2= II-P1;
end;%Solving via CVX:
cvx_begin sdp quiet
variables E1(2,2) E2(2,2);
E1==hermitian_semidefinite(2); E2==hermitian_semidefinite(2);
maximise(real(trace(Z1*E1)+trace(Z2*(E2))));
subject to E1+E2==II; trace(E1)<=1.; trace(E2)<=1.;cvx_end
RHS= real(trace(Z1E1)+trace(Z2E2))
RHS= real(trace(Z1P1)+trace(Z2P2))
The output is:
RHS =
0.0411
RHS =
0.0807
So the minimal solution is not reached by CVX. I found out that E1, E2 are both “real” in the solution. Could that be the reason? Or do I misunderstand CVX somehow?
Thank you for your help and comments!