I have a system of the form $Ax = b$, and I want to obtain the best $x$ in the least squares sense. $A$ is $M \times N$, where $M \approx 2N$ (or $4N$ for another variation). Rows in $A$ have 2 nonzero columns, as my equations are effectively $x_i - x_j = b$.
I also have K modeling constraints, of the form: $x_i = c_i$, $K \ll M$.
My problem is with weighting.
Everything has different importance, the constraints are generally much more important (some of utmost importance), and I don't know how to go about the scale and distribution of weights. Example questions.
1) Should the weights definitely add up to 1?
2) If I have weights adding up to one, is there a 'rule' of some sorts of distributing weights? e.g if $M = 100K$, is there an ideal weight percentage distributed over all $K$ constraints?
3) Is it a good idea to use really small weight values for loosely enforcing some constraints? e.g. $x_i = 0.001c_i$ : I want $x_i$ to be around, not necessarily at $c_i$