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 R-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(a)(x(k)) of x(k) to define a v(k) is an element of partial derivative(ek) f(p(a)(x(k))) with epsilon(k) less than or equal to alpha parallel to v(k)parallel to where alpha is a constant. The method monitors the reduction in the value of parallel to v(k)parallel to to identify when a line search on f should be used. The quasi-Newton step is used to reduce the value of parallel to v(k)parallel to 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 More. Numerical results show the good performance of the method.