A tutorial on decomposition methods for network utility maximization

Abstract

Abstract-A systematic understanding of the decomposability structures in network utility maximization is key to both resource allocation and functionality allocation. It helps us obtain the most appropriate distributed algorithm for a given network resource allocation problem, and quantifies the comparison across architectural alternatives of modularized network design. Decomposition theory naturally provides the mathematical language to build an analytic foundation for the design of modularized and distributed control of networks.In this tutorial paper, we first review the basics of convexity, Lagrange duality, distributed subgradient method, Jacobi and Gauss-Seidel iterations, and implication of different time scales of variable updates. Then, we introduce primal, dual, indirect, partial, and hierarchical decompositions, focusing on network utility maximization problem formulations and the meanings of primal and dual decompositions in terms of network architectures. Finally, we present recent examples on: systematic search for alternative decompositions; decoupling techniques for coupled objective functions; and decoupling techniques for coupled constraint sets that are not readily decomposable.

Centralized versus decentralized multi-cell resource and power allocation for multiuser OFDMA networks

“…Since the power allocation problem (10) is a convex optimization problem, we can make use of Lagrange duality properties, which also lead to decomposability 240 structures [29]. Lagrange duality theory links the original problem, or primal problem, with a dual maximization problem.…”

“…The scalar δ(t) is a step size that guarantees the convergence of the primal optimization problem [29]. The partial derivatives of the objective function L( ρ, π, λ, ν) with respect to ρ k,i,n and π i,n , are given in the following:…”

Optimization of Visible-Light Optical Wireless Systems: Network-Centric Versus User-Centric Designs

“…Broadly speaking, the LB problem can be formulated as the constrained optimization of a carefully selected utility function [137], while satisfying the users' quality of service (QoS) requirements, which may be written as the generic optimization problem formulated below:…”

“…Furthermore, f c i is the ith cost function and C i is the corresponding ith AP/network resource limit, while f t j denotes the jth UEs achievable data throughput and T j corresponds to its throughput requirement. If (14)- (16) are all convex 6 , the problem is a convex optimization problem [137]. However, due to the coupled relationship between the user association and scheduling, the problem of (14)- (16) is usually NP-hard and may not be directly computable in a centralized manner.…”

Fair Resource Allocation with QoS Guarantee in Secure Multiuser TDMA Networks

“…First, we discuss the issue of minimizing a general cost function of the average user power subjected to the delay QoS constraint of the individual user. A dual-based iterative algorithm based on the Lagrangian dual technique [28,29] and the dual decomposition theory [30,31] is employed to solve the optimization problem. In addition, according to the stochastic optimization theory, we develop a stochastic subgradient iterative algorithm with unknown cumulative distribution function (CDF) of the fading channel, which can dynamically learn the underlying channel distribution and approach the optimal solution.…”