Proximal quasi-Newton methods for nondifferentiable convex optimization

This paper proposes an implementable proximal quasi-Newton method for minimizing a nondifferentiable convex function f in <n . The method is based on Rockafellar’s proximal point algorithm and a cutting-plane technique. At each step, we use an approximate proximal point p(xk) of xk to define a vk ∈ ∂ k f(p(xk))with k ≤ α‖vk‖,where α is a constant. The method monitors the reduction in the value of ‖vk‖ to identify when a line search on f should be used. The quasi-Newton step is used to reduce the value of ‖vk‖. Without the differentiability of f , the method converges globally and the rate of convergence is Q-linear. Superlinear convergence is also discussed to extend the characterization result of Dennis and Moré. Numerical results show the good performance of the method.