Investigating Degenerate Preconditioners for Proximal Point Algorithms

In this thesis, we consider and study fixed point iterations to solve monotone inclusion problems. Specifically, we consider proximal point algorithms that make use of preconditioners. We allow the preconditioners to be degenerate, which means that they might have a non-trivial kernel. The introduction of degenerate preconditioners raises various theoretical questions, some of which we address in this work. To establish the convergence of the sequence generated by the degenerate preconditioned proximal point algorithm, we adapt classical analysis to the degenerate case, imposing mild additional assumptions that do not hinder the practical applications of the method. Additionally, we demonstrate that the degeneracy of the preconditioner enables the reduction of the dimension of the space where the algorithm is defined, without loss of information between an iterate and the next. Furthermore, we provide a characterization of the involved operator for this new reduced scheme. In recent years, many classical proximal splitting schemes designed to tackle monotone inclusion problems featuring sums of multiple operators, have been identified as proximal point iterations. For this reason, we show the main application of the framework in the context of splitting algorithms. By incorporating degenerate preconditioners, we can encompass a broader range of splitting schemes, offering new insights and a unified convergence analysis for these methods. Examples of such algorithms include the Douglas- Rachford method, the Peaceman-Rachford method, the Chambolle-Pock algorithm, and the Davis-Yin method. As a further step, we leverage the generality of the framework to propose new generalizations for these splitting schemes. Specifically, we introduce graph extensions for both the Douglas-Rachford method and the Davis-Yin algorithm, studying their behavior, and drawing interesting relations between their performance and the structures of the employed graphs. In the last part of the thesis, we consider possible generalizations of the degenerate preconditioned proximal point algorithm. We first study a proximal point iteration with non-stationary degenerate preconditioners and investigate a specific application to the primal-dual Douglas-Rachford splitting. We thus introduce an inexact version of the algorithm, providing convergence analysis for scenarios where each iteration introduces an error generating a summable sequence, as well as for a novel degenerate inexact method with relative errors. We conclude by analyzing an inertial version of the method, providing some examples and a new restart strategy. These expansions of the algorithm result in generalizations for all the splitting schemes that are encompassed in the framework. We demonstrate the practical applications of these extensions and highlight their advantages.