Proximal point algorithms, dynamical systems, and associated operators: modern perspectives from experimental mathematics

Discrete dynamical systems are ubiquitous in many mathematical disciplines. Celebrated methods as old as the Hellenic period include Euclid’s algorithm and continued fractions for real numbers. In modern convex analysis and nonlinear optimization, they often take on the form — or inspiration — of proximal point algorithms. The modern tools and methods of experimental mathematics have irreversibly changed the mathematical landscape, not only for the analysis of newer algorithms but for reinventions of old ones. One example is the recent introduction of continued logarithms for general bases. This thesis addresses many research questions of interest related to discrete dynamical systems and their associated operators, while giving particular attention to the modern tools and methods employed in the discovery. Its main contributions are threefold. The first is related to the main family of discrete dynamical systems under consideration: those induced by the Douglas–Rachford family of operators and algorithms. We thoroughly describe the history of this family of methods, and we rigorously analyse a collection of important nonconvex cases which are prototypical of more complicated problems to which Douglas–Rachford has been successfully applied in the absence of theoretical justification. The second is to characterise operators related to proximal point algorithms. In particular, we study Bregman envelopes for useful entropy functions, and we describe the convex conjugates for weighted sums and proximal averages of these entropy functions. The third is to furnish a suite of methods adaptable to the investigation of open questions more broadly. To this end, we include many illustrations not found in the original publications on which this body of work is based.