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

This paper 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 feasibil...

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Bibliographic Details
Published in:IEEE transactions on signal processing 2016-04, Vol.64 (7), p.1774-1783
Main Authors: Cavalcante, Renato L. G., Yuxiang Shen, Stanczak, Slawomir
Format: Article
Language:English
Subjects:
Bandwidth
convex functions
Estimation
Interference
interference coupling
Iterative algorithms
nonlinear equations
Optimization
Planning
Scalability
Wireless networks
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Summary:This paper 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 estimation in networks based on the orthogonal frequency division multiple access (OFDMA) technology.
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2015.2502544