Utility Maximization in Nonconvex Wireless Systems
This monograph formulates a framework for modeling and solving utility maximization problems in nonconvex wireless systems. First, a model for utility optimization in wireless systems is defined. The model is general enough to encompass a wide array of sy
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Johannes Brehmer
Utility Maximization in Nonconvex Wireless Systems
123
Johannes Brehmer Technische Universit¨at M¨unchen Associate Institute for Signal Processing 80290 Munich Germany
ISSN 1863-8538 e-ISSN 1863-8546 ISBN 978-3-642-17437-7 e-ISBN 978-3-642-17438-4 DOI 10.1007/978-3-642-17438-4 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2012934555 c Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
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2 General Problem Setup.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Parameter Optimization Problems .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Utility Maximization Problems .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Rate Region and Rate Space Problem .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Proper Rate Regions .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 Intersection Problems . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.3 The Pareto Manifold.. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.4 Outer-Approximation of a Proper Rate Region . . . . . . . . . . . . . . 2.3.5 Convex Rate Regions .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7 7 14 18 20 24 25 29 31
3 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 A Simple Utility Maximization Problem .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Rate Space Formulation .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Lagrange Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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