I am using the R interface to the Lawson-Hanson NNLS implementation of an algorithm for non-negative linear least squares that solves ||A x - b||^2 with the constraint that all elements of vector x ≥ 0. This works fine but I would like to add further constrains. Of interest to me are:
Also minimize "energy" of x:
||A x - b||^2 + m*||x||^2
Minimize "energy in the x derivative"
||A x - b||^2 + m ||H x||^2, where H is the sum of identity and a matrix with -1 on the first off-diagonal
Most generally, minimize ||A x - b||^2 + m ||H x - f||^2.
Is there are a way to coax nnls to do this by some clever way of restating the problems 1.-3. Above? The reason I have hope for such a thing is that there is a little-throw away comment in a paper by Whitall et al (sorry for the paywall) that claims that "fortunately, NNLS can be adopted from the original form above to accommodate something in problem 3".
I take it m is a scalar, right? Consider the simple case m=1; you can generalize for other values of m by letting H* = sqrt(m) H and f* = sqrt(m) f and using the solution method given here.
So now you're trying to minimise ||A x - b||^2 + ||H x - f||^2.
Let A* = [A' | H']' and let b* = [b' | f']' (i.e. stack up A on top of H and b on top of f) and solve the original problem of non-negative linear least squares on ||A* x - b*||^2 with the constraint that all elements of vector x ≥ 0 .