  We study the combination of proximal gradient descent with multigrid for
solving a class of possibly nonsmooth strongly convex optimization problems. We
propose a multigrid proximal gradient method called MGProx, which accelerates
the proximal gradient method by multigrid, based on using hierarchical
information of the optimization problem. MGProx applies a newly introduced
adaptive restriction operator to simplify the Minkowski sum of subdifferentials
of the nondifferentiable objective function across different levels. We provide
a theoretical characterization of MGProx. First we show that the MGProx update
operator exhibits a fixed-point property. Next, we show that the coarse
correction is a descent direction for the fine variable of the original fine
level problem in the general nonsmooth case. Lastly, under some assumptions we
provide the convergence rate for the algorithm. In the numerical tests on the
Elastic Obstacle Problem, which is an example of nonsmooth convex optimization
problem where multigrid method can be applied, we show that MGProx has a faster
convergence speed than competing methods.
