I would like to solve for unknown parameters X using L1 norm minimization of the absolute sum of the residuals in the form AX = b. My question is, since I also have a proper knowledge about my problem, how can I minimize the weighted absolute sum of the residuals by knowing the weight matrix of observations?
I’ll try to make my problem more understandable:
The CVX commands for minimizing the L1 norm of the sum of the residuals are as follows:
cvx_begin variable X(n) minimize( norm(A*X-b,1) ) cvx_end
Now I would like to minimize the weighted sum and I use the following code:
cvx_begin variable X(n) minimize( norm (w'*(A*X-b,1)) ) cvx_end
where w is a vector with the same size as observations and contains the diagonal elements of my observations weight matrix.
I was wondering am I using the correct syntax?
I am getting bad results which are worse that minimizing the residuals without weight.
I am thankful for your answers in advance.