conclusion

Given an adequate threshold $\epsilon$, the Huber problem may be solved using a quasi-Newton solver. The Limited memory BFGS method, a quasi-Newton update, has interesting storage properties that lead to efficient convergence to the local minimum of any convex function. In this paper, I proposed an algorithm to solve the Huber problem using the L-BFGS solver and a MoreThuente line search. This algorithm is then supposed to give a R-linear convergence to the desired solution.