Interference calculus : a general framework for interference management and network utility optimization

Cover......Page 1
Front matter......Page 2
Introduction......Page 11
Notation......Page 13
Basic Axiomatic Framework of Interference \rFunctions......Page 14
Convex and Concave Interference Functions......Page 15
Log-Convex Interference Functions......Page 16
Examples  Interference in Wireless Networks......Page 17
Linear Interference Function......Page 18
Beamforming......Page 19
Receive Strategies......Page 20
The Spectral Radius  Indicator of Feasibility......Page 21
Min-Max Balancing and Feasible Sets......Page 22
Transmit Strategies and Duality......Page 23
Robust Designs......Page 24
Interference Functions in Other Contexts......Page 25
Systems of Coupled Interference Functions......Page 27
Interference Coupling......Page 28
Asymptotic Coupling Matrix......Page 29
The Dependency Matrix......Page 30
Strict Monotonicity and Strict Log-Convexity......Page 31
Standard Interference Functions......Page 33
Comparison between Standard Interference Functions and \rGeneral Interference Functions......Page 34
Continuity......Page 36
Continuation on the Boundary......Page 37
Continuity of Standard Interference Functions......Page 38
SIR-Based QoS Sets......Page 39
Comprehensiveness......Page 40
Fixed Point Characterization of Boundary Points......Page 41
Power-Constrained QoS Regions......Page 42
Sum-Power Constraint......Page 43
Individual Power Constraints......Page 44
The QoS Balancing Problem......Page 45
Equivalent Convex Reformulations......Page 46
Equivalent Log-Convex Reformulation......Page 47
The Structure of Interference Functions and \rComprehensive Sets......Page 49
Max-Min and Min-Max Representations......Page 50
Majorants and Minorants......Page 51
Representation Theorem based on Level Sets......Page 52
Elementary Sets and Interference Functions......Page 54
Interference Functions and Comprehensive Sets......Page 56
Comprehensive Hull......Page 58
Concave Interference Functions......Page 60
Representation of Concave Interference Functions......Page 61
Properties of the Set $N_0(I)$\r......Page 64
Synthesis of Concave Interference Functions......Page 65
Least Concave Majorant and Alternative Representation......Page 67
Representation of Convex Interference Functions......Page 71
Properties of the set $W_0(I)$\r......Page 73
Synthesis of Convex Interference Functions......Page 74
Greatest Convex Minorant and Alternative Representation......Page 76
Expressing Utility Sets as Sub-/Superlevel Sets of \rConvex/Concave Interference Functions......Page 79
Basic Building Blocks of Log-Convex Interference Functions......Page 82
Analysis of Log-Convex Interference Functions......Page 85
Synthesis of Log-Convex Interference Functions......Page 87
Connection with the Kullback-Leibler Distance......Page 88
Every Convex Interference Function is a Log-Convex \rInterference Function......Page 89
Greatest Log-Convex Minorant......Page 90
Application to Standard Interference Functions......Page 92
Convex Weakly Standard Interference Functions......Page 93
Concave Weakly Standard Interference Functions......Page 95
The Structure of Convex and Concave Standard Interference \rFunctions......Page 96
The Structure of Log-Convex Standard Interference \rFunctions......Page 97
Convex/Concave Bounds......Page 99
Least Concave Majorant and Greatest Convex Minorant......Page 101
Comparison of Convex and Log-Convex Minorants......Page 102
Convex and Concave Approximations of SIR Feasible Sets......Page 103
Convex Comprehensive Level Sets......Page 105
The Structure of Interference Functions and \rComprehensive Sets......Page 0
Nash Bargaining and Proportional Fairness for  Log-Convex Utility Sets......Page 109
Nash Bargaining for Strictly Log-Convex Utility Sets......Page 110
The Conventional Nash Bargaining Solution (NBS)......Page 111
Log-Convex Utility Sets......Page 113
Generalized Nash Bargaining for Compact Log-Convex Sets......Page 114
Non-Symmetric Nash Solution and the Conjugate $f_\rI$......Page 118
The SIR Region of Log-Convex InterferenceFunctions......Page 119
Existence of a Fixed Point for Constant W......Page 120
Min-Max and Max-Min Balancing......Page 124
Generalization to Adaptive W......Page 125
Fixed Point Analysis for General Log-Convex Interference \rFunctions......Page 126
Fairness Gap......Page 129
Proportional Fairness - Boundedness, Existence, and \rStrict Log-Convexity......Page 132
Boundedness of the Cost Function......Page 133
Existence of a Proportionally Fair Optimizer......Page 138
Uniqueness of the Solution......Page 140
Equivalence of Nash Bargaining and Proportional Fairness......Page 145
Weighted Utility and Cost Optimization......Page 147
SINR Region under a Total Power Constraint......Page 148
Individual Power Constraints - Pareto Optimality \rand Strict Convexity......Page 150
Properties of Boundary Points......Page 151
Analysis of the Pareto Boundary......Page 155
Concept of Strongly Coupled Users......Page 157
Strict Monotonicity......Page 159
Strict Log-Convexity......Page 161
QoS-Constrained Power Minimization......Page 165
Matrix-Based Iteration......Page 167
Feasibility......Page 168
Monotonicity......Page 170
Lipschitz Continuity......Page 171
Global Convergence and Comparison with the Fixed Point \rIteration......Page 172
Continuously Dierentiable Interference Functions......Page 174
Generalization to Non-Smooth Interference Functions......Page 175
Invertibility......Page 176
Non-Smooth Versions of Newton\'s Method......Page 177
Superlinear Convergence......Page 178
Quadratic Convergence for Degree-2 Semi-Continuous \rInterference Functions......Page 180
Convergence of the Fixed Point Iteration......Page 181
Linear Convergence......Page 182
Geometrical Interpretation for the 2-User Beamforming Case......Page 184
Matrix Iteration for Convex Interference Functions......Page 187
Convergence Analysis......Page 190
Weighted SIR Balancing......Page 193
The Max-Min Optimum......Page 194
Set of Optimal Receive Strategies......Page 195
Existence and Uniqueness of the Optimal Power Allocation......Page 196
Equivalence of Max-Min and Min-Max Balancing......Page 197
Principal Eigenvector (PEV) Iteration......Page 198
Fixed Point Iteration......Page 202
Convergence Behavior of the PEV Iteration......Page 205
Back matter......Page 206