Quantitative convergence analysis of iterated expansive, set-valued mappings
Provides a unified convergence theory for nonconvex optimization algorithms, benefiting researchers in optimization and fixed point theory.
The paper develops a framework for quantitative convergence analysis of Picard iterations of expansive set-valued fixed point mappings, proving local linear convergence for nonconvex cyclic projections, forward-backward, and Douglas-Rachford algorithms. This unifies earlier convex and nonconvex results.
We develop a framework for quantitative convergence analysis of Picard iterations of expansive set-valued fixed point mappings. There are two key components of the analysis. The first is a natural generalization of single-valued averaged mappings to expansive, set-valued mappings that characterizes a type of strong calmness of the fixed point mapping. The second component to this analysis is an extension of the well-established notion of metric subregularity -- or inverse calmness -- of the mapping at fixed points. Convergence of expansive fixed point iterations is proved using these two properties, and quantitative estimates are a natural byproduct of the framework. To demonstrate the application of the theory, we prove for the first time a number of results showing local linear convergence of nonconvex cyclic projections for inconsistent (and consistent) feasibility problems, local linear convergence of the forward-backward algorithm for structured optimization without convexity, strong or otherwise, and local linear convergence of the Douglas--Rachford algorithm for structured nonconvex minimization. This theory includes earlier approaches for known results, convex and nonconvex, as special cases.