Non-Euclidean Enriched Contraction Theory for Monotone Operators and Monotone Dynamical Systems

TL;DR

This paper introduces a non-Euclidean contraction framework for analyzing monotone operators and dynamical systems, enabling larger step sizes and faster convergence in fixed-point iterations and multi-agent systems.

Contribution

It develops the concept of enriched weak contractivity in a non-Euclidean setting, extending the analysis of Krasnoselskij iteration and monotone systems.

Findings

Provides explicit step size bounds for convergence.

Relates weak contractivity to operator monotonicity.

Demonstrates improved algorithms for zero-finding and consensus.

Abstract

We adopt an operator-theoretic perspective to analyze a class of nonlinear fixed-point iterations and discrete-time dynamical systems. Specifically, we study the Krasnoselskij iteration - at the heart of countless algorithmic schemes and underpinning the stability analysis of numerous dynamical models - by focusing on a non-Euclidean vector space equipped with the diagonally weighted supremum norm. By extending the state of the art, we introduce the notion of enriched weak contractivity, which (i) is characterized by a simple, verifiable condition for Lipschitz operators, and (ii) yields explicit bounds on the admissible step size for the Krasnoselskij iteration. Our results relate the notion of weak contractivity with that of monotonicity of operators and dynamical systems and show its generality to design larger step sizes and improved convergence speed for broader classes of dynamical systems. The newly developed theory is illustrated on two applications: the design of zero-finding algorithms for monotone operators and the design of nonlinear consensus dynamics in monotone multi-agent dynamical systems.