Error on quadratic form

Nonconvex
1

Here is my cvx code

cvx_begin
            variable E_user(n) nonnegative
            variable z nonnegative
            expression sigma(n) 
            expression Sigma(n) 
            expression Omega(n) 
            expression A(n)
            expression B(n)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
         
%             sigma=zeros(1,K_num); Sigma=zeros(1,K_num); Omega=zeros(1,K_num);
        
            % Calculate sigma
            for t=1:K_num
                my_hold=zeros(M*N,M*N);
                for i=1:K_num
                    if i~=t
                        my_hold=my_hold+E_user(i)*G_DDMA(:,:,i)*G_DDMA(:,:,i)';
                    end
                end
                sigma(t)=trace(pinv(D(:,:,t))*U(:,:,t)*H(:,:,t)*my_hold*H(:,:,t)'*U(:,:,t)'*(pinv(D(:,:,t)))');
            end
        
            % Calculate Sigma
            for t=2:K_num
                my_hold=zeros(M*N,M*N);
                for i=2:K_num
                    if i~=t
                        my_hold=my_hold+E_user(i)*G_DDMA(:,:,i)*G_DDMA(:,:,i)';
                    end
                end
                Sigma(t)=trace(pinv(D(:,:,1))*U(:,:,1)*H(:,:,1)*my_hold*H(:,:,1)'*U(:,:,1)'*(pinv(D(:,:,1)))');
            end
            my_hold=zeros(M*N,M*N);
            for i=3:K_num
                my_hold=my_hold+E_user(i)*G_DDMA(:,:,i)*G_DDMA(:,:,i)';
            end
            Sigma(1)=trace(pinv(D(:,:,2))*U(:,:,2)*H(:,:,2)*my_hold*H(:,:,2)'*U(:,:,2)'*(pinv(D(:,:,2)))');
        
            % Calculate Omega
            for i=2:K_num
                Omega(i)=trace(pinv(D(:,:,1))*U(:,:,1)*H(:,:,1)*G_DDMA(:,:,i)*G_DDMA(:,:,i)'*H(:,:,1)'*U(:,:,1)'*(pinv(D(:,:,1)))');
            end
            Omega(1)=trace(pinv(D(:,:,2))*U(:,:,2)*H(:,:,2)*G_DDMA(:,:,1)*G_DDMA(:,:,1)'*H(:,:,2)'*U(:,:,2)'*(pinv(D(:,:,2)))');
        
            sigma=real(sigma);
            Sigma=real(Sigma);
            Omega=real(Omega);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
            for i=2:n
                A(i)=(N0+alpha*sigma(i)+E_user(i))*(N0+E_user(1)+alpha*Sigma(1));
                B(i)=(N0+alpha*Sigma(i))*(N0+E_user(1)+alpha*Sigma(1)+E_user(i)*Omega(1));
            end
            A(1)=(N0+alpha*sigma(1)+E_user(1))*(N0+E_user(2)+alpha*Sigma(2));
            B(1)=(N0+alpha*Sigma(1))*(N0+E_user(2)+alpha*Sigma(2)+E_user(1)*Omega(2));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

            maximize (z)
            E_sum=0;
            for i=1:n
                E_sum=E_sum+E_user(i);
            end 
            subject to 
                E_sum<=E_total;
                for i=2:n
                    log(2*beta(i)*sqrt(A(i)) -  beta(i)^2*B(i) )/log(2)>=z;
                end
                log(2*beta(1)*sqrt(A(1)) - beta(1)^2*B(1) )/log(2)>=z;
        cvx_end

I have following error when cvx is going to generate A(i)

Disciplined convex programming error:
    Invalid quadratic form(s): not a square.

I believe the error comes from I have made variables production when generate A(i), is there any method to fix this?

2

Perhaps you can help us out by telling us exactly which statement triggered the error message? If it involves trace, you might be able to use the reformulation
trace (A*A') = square_pos(norm(A,'fro')) , and use cyclic permutation invariance of trace to rearrange the argument of trace to get it in this form if needed.

If that does not handle your situation, your first task is to prove your optimization problem is convex (concave objective being maximized, and all constraints convex, all holding jointly with respect to the CVX variables (accounting for anything going on in the CVX expressions).

3

Thank you for your answer, the error is triggered by this line in my cvx code.

I think the error is because I made variable product…But I have to do this way…

4

That is non-convex. So it would appear that CVX is not the right tool for this problem. Perhaps you can try a local or global (non-convex) nonlinear optimization solver under YALMIP.

5

Thank you so much for your explaination, it makes sense.