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.