Convergence of the Chambolle-Pock Algorithm in the Absence of Monotonicity

The Chambolle-Pock algorithm (CPA), also known as the primal-dual hybrid gradient method (PDHG), has surged in popularity in the last decade due to its success in solving convex/monotone structured problems. This work provides convergence results for problems with varying degrees of (non)monotonicity, quantified through a so-called oblique weak Minty condition on the associated primal-dual operator. Our results reveal novel stepsize and relaxation parameter ranges which do not only depend on the norm of the linear mapping, but also on its other singular values. In particular, in nonmonotone settings, in addition to the classical stepsize conditions for CPA, extra bounds on the stepsizes and relaxation parameters are required. On the other hand, in the strongly monotone setting, the relaxation parameter is allowed to exceed the classical upper bound of two. Moreover, sufficient convergence conditions are obtained when the individual operators belong to the recently introduced class of semimonotone operators. Since this class of operators encompasses many traditional operator classes including (hypo)- and co(hypo)monotone operators, this analysis recovers and extends existing results for CPA. Several examples are provided for the aforementioned problem classes to demonstrate and establish tightness of the proposed stepsize ranges.