Primal-dual multigrid methods for nonsmooth optimization

Abstract

In optimization, one often encounters problems of the form ${\min }_{x}F(x)+E(x)+G(Kx)$. In this work, we combine primal-dual algorithms with multigrid techniques for their solution. To link the the fine-grid and coarse-grid problems problems, we introduce a nonsmooth primal-dual coherence condition, and an efficient partially linearized line search procedure. Our work is motivated by total variation regularized inverse imaging problems, on which we demonstrate the efficacy of the method, being able to solve problems not previously possible with forward-backward multigrid methods.