Proximal Quasi-Newton Methods for Nondifferentiable Convex Optimization

Some global convergence properties of a variable metric algorithm for minimization without exact line searches, in R. 23 superlinear convergent algorithm for minimizing the Moreau-Yosida regularization F. However, this algorithm makes use of the generalized Jacobian of F, instead of matrices B k generated by a quasi-Newton formula. Moreover, the line search is performed on the function F , rather than f, which is usually quite expensive. In [29], a modication of the algorithm of [12] is presented and its local convergence properties are studied. The algorithm proposed in this paper uses a quasi-Newton update of B k and line search is done on the function f. The resulting algorithm is globally and superlinearly convergent under suitable conditions, and it is imple-mentable. Numerical results show its good performance for piecewise quadratic optimization problems. Acknowledgements. We are grateful to J. Womersley and two referees for their helpful comments. We thank K. Kiwiel for providing his FORTRAN NDO test problems and related references. On the local super-linear convergence of a matrix secant implementation of the variable metric proximal point algorithm, On the superlinear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating,