A $\operatorname{prox}$-Based Semi-Smooth Newton Method for Convex Variational Problems

Abstract

In this paper, we devise a $\operatorname{prox}$-based semi-smooth Newton method that is applicable to a finite element discretization of a broad class of nonsmooth convex variational problems, including the TV-minimization problem, the $p$-Dirichlet problem, the obstacle problem, and the elasto-plastic torsion problem. To this end, on the basis of the proximity operator, the discrete primal-dual optimality conditions are reformulated as nonlinear operator equations with Newton-differentiable structure. Under suitable assumptions on the energy densities, we establish the global well-posedness and local super-linear convergence of the resulting semi-smooth Newton method. The proposed approach coincides with established semi-smooth Newton methods for obstacle-type problems, satisfies a primal-dual invariance, and, under suitable additional assumptions, is globally well-posed in the infinite-dimensional setting.