  The Primal-Dual hybrid gradient (PDHG) method is a powerful optimization
scheme that breaks complex problems into simple sub-steps. Unfortunately, PDHG
methods require the user to choose stepsize parameters, and the speed of
convergence is highly sensitive to this choice. We introduce new adaptive PDHG
schemes that automatically tune the stepsize parameters for fast convergence
without user inputs. We prove rigorous convergence results for our methods, and
identify the conditions required for convergence. We also develop practical
implementations of adaptive schemes that formally satisfy the convergence
requirements. Numerical experiments show that adaptive PDHG methods have
advantages over non-adaptive implementations in terms of both efficiency and
simplicity for the user.
