Spingarn's Method and Progressive Decoupling Beyond Elicitable Monotonicity

Spingarn's method of partial inverses and the progressive decoupling algorithm address inclusion problems involving the sum of an operator and the normal cone of a linear subspace, known as linkage problems. Despite their success, existing convergence results are limited to the so-called elicitable monotone setting, where nonmonotonicity is allowed only on the orthogonal complement of the linkage subspace. In this paper, we introduce progressive decoupling+, a generalized version of standard progressive decoupling that incorporates separate relaxation parameters for the linkage subspace and its orthogonal complement. We prove convergence under conditions that link the relaxation parameters to the nonmonotonicity of their respective subspaces and show that the special cases of Spingarn's method and standard progressive decoupling also extend beyond the elicitable monotone setting. Our analysis hinges upon an equivalence between progressive decoupling+ and the preconditioned proximal point algorithm, for which we develop a general local convergence analysis in a certain nonmonotone setting.

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.