Sparse constrained linear least-squares solver

Question

This great SO answer points to a good sparse solver for Ax=b, but I've got constraints on x such that each element in x is >=0 an <=N.

Also, A is huge (around 2e6x2e6) but very sparse with <=4 elements per row.

Any ideas/recommendations? I'm looking for something like MATLAB's lsqlin but with huge sparse matrices.

I'm essentially trying to solve the large scale bounded variable least squares problem on sparse matrices:

EDIT: In CVX:

cvx_begin
    variable x(n)
    minimize( norm(A*x-b) );
    subject to 
        x <= N;
        x >= 0;
cvx_end

Answer 1

Your problem is similar to a nonnegative least-squares problem (NNLS), which can be formulated as

$$\min_x ||Ax-b||_2^2 \text{ subject to } x \ge 0$$,

for which there seems to exist many algorithms.

Actually, you problem can be more or less converted into an NNLS problem, if, in addition to your original nonnegative variables $x$ you create additional variables $x'$ and link them with linear constraints $x_i+x_i'=N$. The problem with this approach is that these additional linear constraints might not be satisfied exactly in the least-squares solution - it might be appropriate then to try to weight them with a large number.

Answer 2

You are trying to solve least squares with box constraints. Standard sparse least squares algorithms include LSQR and more recently, LSMR. These only require you to apply matrix-vector products. To add in the constraints, realize that if you are in the interior of the box (none of the constraints are "active"), then you proceed with whatever interior point method you chose. For all active constraints, the next iteration you perform will either deactivate the constraint, or constrain you to move along the constraint hyperplane. With some (conceptually relatively simple) suitable modifications to the algorithm you choose, you can implement these constraints.

Generally however, you can use any convex optimization package. I have personally solved this exact type of problem using the Matlab package CVX, which uses SDPT3/SeDuMi for a backend. CVX is merely a very convenient wrapper around these semidefinite program solvers.

Answer 3

If you reformulate your model as:

min
subject to:

then it is a standard quadratic programming problem. This is a common type of model that can be easily solved with a commercial solver such as CPLEX or Gurobi. (Disclaimer: I currently work for Gurobi Optimization and formerly worked for ILOG, which provided CPLEX).

Answer 4

Your matrix A^T A is positive semi-definite, so your problem is convex; be sure to take advantage of that when setting up your solver.

Most go-to QP solvers are in Fortran and/or non-free; however I've heard good things about OOQP (http://www.mcs.anl.gov/research/projects/otc/Tools/OOQP/OoqpRequestForm.html), though it's a bit of a pain getting a copy.

Answer 5

How about CVXOPT? It works with sparse matrices, and it seems that some of the cone programming solvers may help:

http://abel.ee.ucla.edu/cvxopt/userguide/coneprog.html#quadratic-cone-programs

This is a simple modification of the code in the doc above, to solve your problem:

from cvxopt import matrix, solvers
A = matrix([ [ .3, -.4,  -.2,  -.4,  1.3 ],
             [ .6, 1.2, -1.7,   .3,  -.3 ],
             [-.3,  .0,   .6, -1.2, -2.0 ] ])
b = matrix([ 1.5, .0, -1.2, -.7, .0])
N = 2;
m, n = A.size
I = matrix(0.0, (n,n))
I[::n+1] = 1.0
G = matrix([-I, I])
h = matrix(n*[0.0]  + n*[N])
print G
print h
dims = {'l': n, 'q': [n+1], 's': []}
x = solvers.coneqp(A.T*A, -A.T*b, G, h)['x']#, dims)['x']
print x

CVXOPT support sparse matrices, so it be useful for you.

Answer 6

If you have Matlab, this is something you can do with TFOCS. This is the syntax you would use to solve the problem:

x = tfocs( smooth_quad, { A, -b }, proj_box( 0, N ) );

You can pass A as a function handle if it's too big to fit into memory.