How to solve this problem in CVX?

1

I want to solve a problem like this
image

Here is my cvx code

cvx_begin sdp
variable W_k(para.M,para.M*para.K) complex
variable Wk(para.M,para.M) complex
expression t
sum=cvx(zeros(para.M,para.M));
tr_sum=0;

       minimize(-t)
          subject to      
          for i=1:para.K
              Wk=W_k(:,para.M*(i-1)+1:para.M*i);
              Wk_sum=sum+Wk;
              tr_sum=tr_sum+trace(Wk);
              Wk>=0;
          end
          [trace(A1'*A1*Wk_sum)-t   trace(A1'*A*Wk_sum);...
                                              trace(A'*A1*Wk_sum)   trace(A'*A*Wk_sum)]==hermitian_semidefinite(2);
           tr_sum<=para.Pt; 
           for k=1:(para.K)
               h=H(:,k);
               ttr=0;
               for f=1:para.K
                   if f ~= k
                       Wk=W_k(:,para.M*(f-1)+1:para.M*f);
                       ttr=ttr+trace(Wk*h*h');
                   end
               end
               Wk=W_k(:,para.M*(k-1)+1:para.M*k);
               trace(Wk*h*h')-gamma*ttr>=gamma*para.sigma_k;
           end
    cvx_end

Here is error information.
image

2

try real(tr_sum)<=para.Pt;

3

You need to declare W as complex hermitian. in order for W >= 0 to be treated as a smidefinite constraint. Or just declare W as hermitian semidefinite, and then you don’t even need W >= 0.

Perhaps that will make your already encountered error go away, even without the real(). Or maybe placing parentheses around A1'*A which couldn’t hurt., and might help. You can use the real() if you want to.

4

Thank you for your help.Then there is new problem.
image

5

As it is now, t is an expression, which by default, until overriden, has value zero. So you are minimizing -0, and the t in the constraint has value 0.

The warning is because Wk is declared as a variable, but you then assign it, and hence it becomes an expression rather than variable. I suggest you declare

variable t
variable W_k(para.M,para.M*para.K) hermitian semidefinite

now you don’t need to any additional semidefinite constraints for W_k.

Then use expressions such as
trace((A'*A)*sum(W_k,3))-t which is the upper left block of the block semidefinite constraint. You can do that instead of the complicated indexing mess you now have.

@jackfsuia might have a clever vectorized way of writing the middle constraints. But you can always use a for loop, without your messy indexing.

6

Thank you for your help!