The Chambolle-Pock algorithm (CPA), also known as the primal-dual hybrid gradient method, has gained popularity over the last decade due to its success in solving large-scale convex structured problems. This work extends its convergence analysis 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, 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, we build upon the recently introduced class of semimonotone operators, providing sufficient convergence conditions for CPA when the individual operators are semimonotone. Since this class of operators encompasses traditional operator classes including (hypo)- and co(hypo)-monotone operators, this analysis recovers and extends existing results for CPA. Tightness of the proposed stepsize ranges is demonstrated through several examples.

翻译：Chambolle-Pock算法（CPA），亦称为原始-对偶混合梯度法，因其在求解大规模凸结构化问题上的成功，在过去十年中广受欢迎。本文将其收敛性分析推广至具有不同程度（非）单调性的问题，这种单调性通过关联原始-对偶算子的所谓斜弱Minty条件进行量化。我们的结果揭示了新颖的步长和松弛参数范围，这些范围不仅依赖于线性映射的范数，还依赖于其其他奇异值。特别地，在非单调设置下，除了经典的步长条件外，还需要对步长和松弛参数施加额外的界限。另一方面，在强单调设置下，松弛参数被允许超过经典的上界2。此外，我们基于最近引入的半单调算子类，为当各算子为半单调时CPA的收敛提供了充分条件。由于此类算子涵盖了包括（次）单调和共（次）单调算子在内的传统算子类，该分析恢复并扩展了CPA的现有结果。通过多个示例，证明了所提步长范围的紧致性。