The fixed point iteration of positive concave mappings converges geometrically if a fixed point exists: implications to wireless systems

TL;DR

This paper proves that fixed point iterations of positive concave mappings with existing fixed points converge geometrically, explaining the observed rapid convergence of certain wireless network algorithms.

Contribution

It establishes geometric convergence of fixed point iterations for positive concave mappings and highlights the importance of positivity for this property.

Findings

Fixed point iteration converges geometrically if a fixed point exists.

Positivity of the mapping is essential for geometric convergence.

Results explain rapid convergence in wireless power control algorithms.

Abstract

We prove that the fixed point iteration of arbitrary positive concave mappings with nonempty fixed point set converges geometrically for any starting point. We also show that positivity is crucial for this result to hold, and the concept of (nonlinear) spectral radius of asymptotic mappings provides us with information about the convergence factor. As a practical implication of the results shown here, we rigorously explain why some power control and load estimation algorithms in wireless networks, which are particular instances of the fixed point iteration, have shown geometric convergence in simulations. These algorithms have been typically derived by considering fixed point iterations of the general class of standard interference mappings, so the possibility of sublinear convergence rate could not be ruled out in previous studies, except in special cases that are often more restrictive than those considered here.