Sparse constrained linear least-squares solver?

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.

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

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