A trust region method for nonsmooth convex optimization

We propose an iterative method that solves a nonsmooth convex optimization problem by converting the original objective function to a once continuously differentiable function by way of Moreau-Yosida regularization. The proposed method makes use of approximate function and gradient values of the Moreau-Yosida regularization instead of the corresponding exact values. Under this setting, Fukushima and Qi (1996) and Rauf and Fukushima (2000) proposed a proximal Newton method and a proximal BFGS method, respectively, for nonsmooth convex optimization. While these methods employ a line search strategy to achieve global convergence, the method proposed in this paper uses a trust region strategy. We establish global and superlinear convergence of the method under appropriate assumptions.