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Asked: 2026-05-15T09:01:43+00:00

This great SO answer points to a good sparse solver for Ax=b , but

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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:

alt text

EDIT:
In CVX:

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

  1. Editorial Team

    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.

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