  We leverage the connections between nonexpansive maps, monotone Lipschitz
operators, and proximal mappings to obtain near-optimal (i.e., optimal up to
poly-log factors in terms of iteration complexity) and parameter-free methods
for solving monotone inclusion problems. These results immediately translate
into near-optimal guarantees for approximating strong solutions to variational
inequality problems, approximating convex-concave min-max optimization
problems, and minimizing the norm of the gradient in min-max optimization
problems. Our analysis is based on a novel and simple potential-based proof of
convergence of Halpern iteration, a classical iteration for finding fixed
points of nonexpansive maps. Additionally, we provide a series of algorithmic
reductions that highlight connections between different problem classes and
lead to lower bounds that certify near-optimality of the studied methods.
