The linear e1 estimator and the Huber m-estimator

Relationships between a linear l(1) estimation problem and the Huber M-estimator problem can be easily established by their dual formulations. The least norm solution of a linear programming problem studied by Mangasarian and Meyer [SIAM J. Control Optim., 17 (1979), pp. 745-752] provides a key link between the dual problems. Based on the dual formulations, we establish a local linearity property of the Huber M-estimators with respect to the tuning parameter and prove that the solution set of the Huber M-estimator problem is Lipschitz continuous with respect to perturbations of the tuning parameter. As a consequence, the set of the linear l(1) estimators is the limit of the set of the Huber M-estimators as --> 0(+). Thus, the Huber M-estimator problem has many solutions for small tuning parameter if the linear l(1) estimation problem has multiple solutions. A recursive version of Madsen and Nielsen's algorithm [SIAM J. Optim., 3 (1993), pp. 223-235] based on computation of the Huber M-estimator is proposed for finding a linear l(1) estimator.