  We present several algorithms aimed at constructing sparse and structured
sparse (row-sparse) generalized inverses, with application to the efficient
computation of least-squares solutions, for inconsistent systems of linear
equations, in the setting of multiple right-hand sides and a rank-deficient
constraint matrix. Leveraging our earlier formulations to minimize the 1- and
2,1- norms of generalized inverses that satisfy important properties of the
Moore-Penrose pseudoinverse, we develop efficient and scalable ADMM algorithms
to address these norm-minimization problems and to limit the number of nonzero
rows in the solution. We establish a 2,1-norm approximation result for a
local-search procedure that was originally designed for 1-norm minimization,
and we compare the ADMM algorithms with the local-search procedure and with
general-purpose optimization solvers.
