Scaled Relative Graph: Nonexpansive operators via 2D Euclidean Geometry

Abstract

Many iterative methods in applied mathematics can be thought of as fixed-point iterations, and such algorithms are usually analyzed analytically, with inequalities. In this paper, we present a geometric approach to analyzing contractive and nonexpansive fixed point iterations with a new tool called the scaled relative graph (SRG). The SRG provides a correspondence between nonlinear operators and subsets of the 2D plane. Under this framework, a geometric argument in the 2D plane becomes a rigorous proof of convergence.

Published in Mathematical Programming

Cited by in corpus (5)

• Graphical Nonlinear System Analysis
• Monotone one-port circuits
• A rolled-off passivity theorem
• Scaled Relative Graphs for Nonmonotone Operators with Applications in Circuit Theory
• Convergence Analyses of Davis-Yin Splitting via Scaled Relative Graphs II: Convex Optimization Problems