Fixed Point Algorithms for Nonconvex Feasibility with Applications

Abstract

Projection algorithms for solving (nonconvex) feasibility problems provide powerful and computationally efficient schemes for a wide variety of applications. Algorithms as Alternating Projections (AP) and Douglas-Rachford (DR) are two of the more prominent projection algorithms in imaging sciences and signal processing. A nonconvex framework is introduced that enables a general and new approach to characterizing the convergence behavior of general fixed point operators. In classical fixed point theory, firm nonexpansiveness of mappings is a property that is often used to show convergence of a broad class of algorithms. As our main interest is dealing with nonconvex feasibility, the described methods no longer match the notion of firm nonexpansiveness. The framework, theorems and concepts developed in this work generalize the tools from convex analysis for the analysis of fixed-point iterations of operators that violate the classical property of firm nonexpansiveness in some quantifiable fashion. Using these techniques a convergence analysis on AP and DR is carried out. The analysis is then applied to the problem of phase retrieval and ptychographic imaging.