Linear matrix inequalities results in zeros solution


(Daniel Pérez) #1

Hello,

I’m trying to convert the previous semidefinite problem into cvx code. Here is the code I am using:
%%
cvx_begin quiet
variable N(m+1,m+1) hermitian semidefinite
variable D(m+1,m+1) hermitian semidefinite
variable X1(m+1,m+1) hermitian semidefinite
variable X2(m,m) hermitian semidefinite
variable X3(m+1,m+1) hermitian semidefinite
variable X4(m,m) hermitian semidefinite
variable X5(m+1,m+1) hermitian semidefinite
variable X6(m,m) hermitian semidefinite

% passband constraints

    (diag(W1*((N-D)+X1)*W1')+Rp*diag(W2*X2*W2'))==0;

    (diag(W1*((delta2p*D-N)+X3)*W1')+Rp*diag(W2*X4*W2'))==0;
% stopband constraint
    (diag(W1*((N-delta2s*D)+X5)*W1')+Rs*diag(W2*X6*W2'))==0;

cvx_end

I am not sure if I have done the translation to cvx properly. Doing in this way I obtain
All zero sparse: 5-by-5 on every variable.

Thank you in advance,


(Mark L. Stone) #2

The matrices of all zeros are a feasible solution. You have not provided an objective function, so CVX and its solvers only endeavor to produce a feasible solution, which they have. Is this supposed to be a feasibility problem, or is it an optimization problem, in which case you should supply an objective function.

Your model description says D is positive definite, which would exclude the all zeros solution. But you have declared D as hermitian semidefinite, which means a semidefinite solution is allowed, not required to be strictly positive definite. If this distinction is crucial to your problem, you will need to do something about it - perhaps something like putting a lower bound on lambda_min(D), such as perhaps 1e-6 or so. Perhaps some other fix is appropriate.