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The primal--dual hybrid gradient method (PDHGM, also known as theChambolle--Pock method) has proved very successful for convex optimizationproblems involving linear operators arising in image processing and inverseproblems. In this paper, we analyze an extension to nonconvex problems thatarise if the operator is nonlinear. Based on the idea of testing, we derive newstep length parameter conditions for the convergence in infinite-dimensionalHilbert spaces and provide acceleration rules for suitably (locally and/orpartially) monotone problems. Importantly, we prove linear convergence rates aswell as global convergence in certain cases. We demonstrate the efficacy ofthese step length rules for PDE-constrained optimization problems.

Acceleration and global convergence of a first-order primal--dual method for nonconvex problems