This paper investigates the asymptotic and non-asymptotic behavior of the quantized primal dual algorithm in network utility maximization problems, in which a group of agents maximize the sum of their individual concave objective functions under linear constraints. In the asymptotic scenario, we use the information theoretic notion of differential entropy power to establish universal lower bounds on the exponential convergence rates of joint primal dual, primal and dual variables under optimum achieving quantization schemes. These results provide trade offs between the speed of exponential convergence, the agents objective functions, the communication bit rates, and the number of agents and constraints. In the non-asymptotic scenario, we obtain lower bounds on the mean square distance of joint primal dual, primal and dual variables from the optimal solution for any finite time instance. These bounds hold regardless of the quantization scheme used.
Although the performance of distributed optimization algorithms, and in particular NUM algorithms, under perfect communication networks is well understood, the investigation of the impact of imperfect communications on these optimization algorithms is relatively a new research area that has attracted much interests in recent years, see @cite_12 - @cite_8 .
@cite_12 @cite_8
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The standard Network Utility Maximization (NUM) problem has a static formulation, which fails to capture the temporal dynamics in modern networks. This work considers a dynamic version of the NUM problem by introducing additional constraints, referred to as delivery contracts. Each delivery contract specifies the amount of information that needs to be delivered over a certain time interval for a particular source and is motivated by applications such as video streaming or webpage loading. The existing distributed algorithms for the Network Utility Maximization problems are either only applicable for the static version of the problem or rely on dual decomposition and first-order (gradient or subgradient) methods, which are slow in convergence. In this work, we develop a distributed Newton-type algorithm for the dynamic problem, which is implemented in the primal space and involves computing the dual variables at each primal step. We propose a novel distributed iterative approach for calculating the dual variables with finite termination based on matrix splitting techniques. It can be shown that if the error level in the Newton direction (resulting from finite termination of dual iterations) is below a certain threshold, then the algorithm achieves local quadratic convergence rate to an error neighborhood of the optimal solution in the primal space. Simulation results demonstrate significant convergence rate improvement of our algorithm, relative to the existing first-order methods based on dual decomposition. Distributed network utility maximization (NUM) has received an increasing intensity of interest over the past few years. Distributed solutions (e.g., the primal-dual gradient method) have been intensively investigated under fading channels. As such distributed solutions involve iterative updating and explicit message passing, it is unrealistic to assume that the wireless channel remains unchanged during the iterations. Unfortunately, the behavior of those distributed solutions under time-varying channels is in general unknown. In this paper, we shall investigate the convergence behavior and tracking errors of the iterative primal-dual scaled gradient algorithm (PDSGA) with dynamic scaling matrices (DSC) for solving distributive NUM problems under time-varying fading channels. We shall also study a specific application example, namely the multicommodity flow control and multicarrier power allocation problem in multihop ad hoc networks. Our analysis shows that the PDSGA converges to a limit region rather than a single point under the finite state Markov chain (FSMC) fading channels. We also show that the order of growth of the tracking errors is given by O(T N), where T and N are the update interval and the average sojourn time of the FSMC, respectively. Based on this analysis, we derive a low complexity distributive adaptation algorithm for determining the adaptive scaling matrices, which can be implemented distributively at each transmitter. The numerical results show the superior performance of the proposed dynamic scaling matrix algorithm over several baseline schemes, such as the regular primal-dual gradient algorithm. Allocating limited resources such as bandwidth and power in a multi-hop wireless network can be formulated as a Network Utility Maximization (NUM) problem. In this approach, both transmitting source nodes and relaying link nodes exchange information allowing for the NUM problem to be solved in an iterative distributed manner. Some previous NUM formulations of wireless network problems have considered the parameters of data rate, reliability, and transmitter powers either in the source utility function which measures an application's performance or as constraints. However, delay is also an important factor in the performance of many applications. In this paper, we consider an additional constraint based on the average queueing delay requirements of the sources. In particular, we examine an augmented NUM formulation in which rate and power control in a wireless network are balanced to achieve bounded average queueing delays for sources. With the additional delay constraints, the augmented NUM problem is non-convex. Therefore, we present a change of variable to transform the problem to a convex problem and we develop a solution which results in a distributed rate and power control algorithm tailored to achieving bounded average queueing delays. Simulation results demonstrate the efficacy of the distributed algorithm. We consider a wireless OFDMA cellular network and address the problem of jointly allocating power to frequencies (subbands) and assigning users to cells, for a variety of elastic and inelastic services. The goal is to maximize the sum of the users' throughput utility functions subject to various constraints, such as minimum throughput requirements. The problem naturally fits into a Network Utility Maximization (NUM) framework with a mixture of concave (e.g., data rates) and nonconcave (e.g., voice video streaming) utilities. The hardness of this nonconvex, mixed integer program prohibits the use of standard convex optimization algorithms, or efficient combinatorial approximation techniques. We devise a randomized algorithm for the said NUM problem, whose proof of asymptotic optimality is derived from the classical framework of interacting particle systems, via a judiciously selected neighborhood structure. The proposed algorithm is highly distributed, asynchronous, requires limited computational effort per node iteration, and yields provable convergence in the limit. Several numerical experiments are presented to illustrate the convergence speed and performance of the proposed method.