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Solution in the L_1 norm, with L_1 regularization, of a linear system Ax=b.

argmin_x  λ||x||_1 + ||Ax-b||_1

Solution in the L_1 norm, with L_1 regularization, of a system of linear inequalities Ax>=b.

argmin_x  λ||x||_1 + ||step(b-Ax)||_1

L_1 solution of overconstrained system Ax=b.

argmin_x ||Ax-b||_1

L_inf solution of overconstrained system Ax=b.

argmin_x ||Ax-b||_inf

L1 solution of underconstrained linear system Ax=b.

argmin_x ||x||_1 such that Ax=b