A preconditioning proximal Newton method for nondifferentiable convex optimization

We propose a proximal Newton method for solving nondiieren-tiable convex optimization. This method combines the generalized Newton method with Rockafellar's proximal point algorithm. At each step, the proximal point is found approximately and the regu-larization matrix is preconditioned to overcome inexactness of this approximation. We show that such a preconditioning is possible within some accuracy and the second-order diierentiability properties of the Moreau-Yosida regularization are invariant with respect to this preconditioning. Based upon these, superlinear convergence is established under a semismoothness condition.