I have an equation of the form $Ax = b$ where $A$ has dimensions 87 by 66 and rank 60. The last 33 rows of my $A$ encode symmetries of the form $x_1 = x_2$. I know I can calculate a least-squares solution, as well as a non-negative least squares solution (albeit mainly through the knowledge of the existence of MATLAB functions that will do this), but want to know if I can penalise certain unknowns such that the end effect is that they obtain values that are horrifically wrong as long as my other unknowns match up better. In particular, the last six of my unknowns are not very important and so it would be great if there was a way that I could allow the solver to ignore (or weight down) their errors, and minimise the errors for the remaining unknowns. Is this possible?
This question deals with the case of penalising equations, but I haven't found a way of penalising unknowns.