Spectral radii of asymptotic mappings and the convergence speed of the standard fixed point algorithm

TL;DR

This paper introduces a spectral radius-based approach to identify and analyze contractive mappings in fixed point algorithms, providing insights into convergence rates and error bounds, especially in wireless network applications.

Contribution

It demonstrates that spectral radii of asymptotic mappings can identify contractive mappings and estimate their contraction moduli, improving convergence analysis methods.

Findings

Spectral radii can identify contractive mappings.

Spectral radii provide lower bounds for estimation errors.

Load estimation algorithms slow down with increased traffic.

Abstract

Important problems in wireless networks can often be solved by computing fixed points of standard or contractive interference mappings, and the conventional fixed point algorithm is widely used for this purpose. Knowing that the mapping used in the algorithm is not only standard but also contractive (or only contractive) is valuable information because we obtain a guarantee of geometric convergence rate, and the rate is related to a property of the mapping called modulus of contraction. To date, contractive mappings and their moduli of contraction have been identified with case-by-case approaches that can be difficult to generalize. To address this limitation of existing approaches, we show in this study that the spectral radii of asymptotic mappings can be used to identify an important subclass of contractive mappings and also to estimate their moduli of contraction. In addition, if the fixed point algorithm is applied to compute fixed points of positive concave mappings, we show that the spectral radii of asymptotic mappings provide us with simple lower bounds for the estimation error of the iterates. An immediate application of this result proves that a known algorithm for load estimation in wireless networks becomes slower with increasing traffic.