Elementary Properties of Positive Concave Mappings with Applications to Network Planning and Optimization

TL;DR

This paper introduces new methods for computing fixed points of positive concave mappings, which are crucial for network planning and optimization, demonstrating faster convergence and applications in wireless network design.

Contribution

It presents two equivalent methods to construct matrices with spectral radius less than one for fixed point existence, and a new mapping that accelerates fixed point convergence.

Findings

Faster convergence of fixed point iterations with the new mapping.

Effective evaluation of network feasibility through fixed point computation.

Application to power and load planning in OFDMA networks.

Abstract

This study presents novel methods for computing fixed points of positive concave mappings and for characterizing the existence of fixed points. These methods are particularly important in planning and optimization tasks in wireless networks. For example, previous studies have shown that the feasibility of a network design can be quickly evaluated by computing the fixed point of a concave mapping that is constructed based on many environmental and network control parameters such as the position of base stations, channel conditions, and antenna tilts. To address this and more general problems, given a positive concave mapping, we show two alternative but equivalent ways to construct a matrix that is guaranteed to have spectral radius strictly smaller than one if the mapping has a fixed point. This matrix is then used to build a new mapping that preserves the fixed point of the original positive concave mapping. We show that the standard fixed point iterations using the new mapping converges faster than the standard iterations applied to the original concave mapping. As exemplary applications of the proposed methods, we consider the problems of power and load planning in networks based on the orthogonal frequency division multiple access (OFDMA) technology.