Duality in robust linear regression using Huberʹs M-estimator Original Research Article

The robust linear regression problem using Huberʹs piecewise-quadratic M-estimator function is considered. Without exception, computational algorithms for this problem have been primal in nature. In this note, a dual formulation of this problem is derived using Lagrangean duality. It is shown that the dual problem is a strictly convex separable quadratic minimization problem with linear equality and box constraints. Furthermore, the primal solution (Huberʹs M-estimate) is obtained as the optimal values of the Lagrange multipliers associated with the dual problem. As a result, Huberʹs M-estimate can be computed using off-the-shelf optimization software.