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Price-Based Distributed Optimization in Large-Scale Networked Systems Price-Based Distributed Optimization in Large-Scale Networked Systems Dissertation document submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial fulfillment of the requirements of degree of Doctor of Philosophy in the School of Dynamic Systems College of Engineering and Applied Sciences by Baisravan HomChaudhuri Summer 2013 MS, University of Cincinnati, 2010 BE, Jadavpur University, 2007 Committee Chair: Dr. Manish Kumar c 2013 Baisravan HomChaudhuri, All Rights Reserved Abstract This work is intended towards the development of distributed optimization methods for large-scale complex networked systems. The advancement in tech- nological fields such as networking, communication and computing has facili- tated the development of networks which are massively large-scale in nature. The examples of such systems include power grid, communication network, Internet, large manufacturing plants, and cloud computing clusters. One of the important challenges in these networked systems is the evaluation of the optimal point of operation of the system. Most of the centralized optimiza- tion algorithms suffer from the curse of dimensionality that raises the issue of scalability of algorithms for large-scale systems. The problem is essentially challenging not only due to high-dimensionality of the problem, but also due to distributed nature of resources, lack of global information and uncertain and dynamic nature of operation of most of these systems. The inadequacies of the traditional centralized optimization techniques in addressing these issues have prompted the researchers to investigate distributed optimization tech- niques. In particular, distributed resource allocation is a promising paradigm of special relevance to complex networked systems. This research work focuses on developing techniques to carry out the global optimization in a distributed fashion that explores the fundamental idea of decomposing the overall opti- mization problem into a number of sub-problems that utilize limited informa- tion exchanged over the network using neighborhood relationships. Inspired by price-based mechanisms, the research develops two methods. First, a dis- tributed optimization method consisting of dual decomposition and update of dual variables in the subgradient direction, also known as market-based meth- ods, is developed for some different classes of resource allocation problems. Although the dual decomposition based subgradient method requires lesser communication and is easy to implement, it has its own drawbacks includ- ing the rate of convergence. To address some of the drawbacks in the field of distributed optimization, in this dissertation, a Newton based distributed interior point optimization method is developed. The proposed Newton based distributed interior point approach, which is iterative in nature, focuses on the generation of both primal and dual feasible solutions at each iteration and development of mechanisms that demand lesser communication. The conver- gence and rate of convergence of both the primal and the dual variables in the system is also analyzed using a benchmark Network Utility Maximization (NUM) problem followed by numerical simulation results for the NUM prob- lem. A comparative study between the proposed distributed and centralized method of optimization is also provided to evaluate the performance of the proposed method. The proposed distributed optimization techniques have been applied to real world systems such as optimal power allocation in Smart Grid and util- ity maximization in Cloud Computing systems. Both the problems belong to the class of large-scale complex network problems and have immense signifi- cance in current and future power distribution and computing world. In the power grids, the power from the generation units is required to be allocated to the end users in an optimal fashion. The challenges in this problem are augmented with the nature of the decision variables, coupling effect in the network, the global constraints in the system, uncertain nature of renewable power generators, and last and not-the-least the large-scale distributed nature of the problem. In cloud computing, resources such as memory, processing, and bandwidth are needed to be allocated to a large number of users to max- imize the users’ quality of experience. Finally, the research focuses on the development of a stochastic distributed optimization methods for solving problems with multi-modal cost functions. As opposed to the unimodal function optimization, the widely practiced gra- dient descent methods fail to reach the global optimum solution when multi- modal cost functions are considered. In this dissertation, an effort is be made to develop a stochastic distributed optimization method that exploits noise based solution update to prevent the algorithm from converging into local optimum solutions. The method is applied to the Network Utility Maximiza- tion problem with multi-modal cost functions, and is compared with Genetic Algorithm (GA). Acknowledgments I would like to express my deep gratitude to Dr. Manish Kumar for his valu- able and constructive suggestions during the planning and development of this research work. His patience and willingness to give his time so generously has been very much appreciated. I have been extremely fortunate to have an ad- viser like Dr. Kumar who gave me the freedom to explore on my own, at the same time guided me when required. My deepest gratitude to Dr. Kelly Cohen, Dr. David Thompson and Dr. Sam Anand for being a part of my PhD committee. I deeply appreciate their insightful comments on my research as they helped me shape my dissertation. I am thankful to my parents and my elder sister. They were always sup- portive to my decisions and always encouragement me to pursue my studies. I would also like to thank my lab mates Alireza Nemati, Balaji Sharma, Benson Isaac, Gaurav Mukherjee, Jiankun Fan, Joseph Anthony Pietrykowski, Paul Rael, Ruoyu Tan and Sushil Garg for always keeping a lively atmosphere in the lab. I am grateful to the National Science Foundation Grant number EFRI- 1024608 for financially supporting me for sometime in my PhD. I am thankful to Dr. Vijay Kumar Devabhaktuni of University of Toledo, Dr. Prasad Calyam of University of Missouri and Kshitij Fadnis of Ohio State University for col- laborating in different projects. Finally, I would like to thank the University of Cincinnati and the School of Dynamic Systems for giving me the opportunity to pursue my PhD degree. Contents 1 Introduction and Motivation 1 1.1NetworkFlowProblem...................... 7 1.2 Coordination in Networked Systems . ............. 8 1.3 Distributed Optimization ..................... 9 1.4 Optimization with Multimodal Functions ............ 11 1.5 Application Areas ......................... 13 1.6 Document outlines ........................ 15 2 Literature Review 17 2.1DecompositionMethods..................... 18 2.2 Distributed Optimization ..................... 19 2.3Multi-modalCostFunction.................... 28 2.4 Application Areas ......................... 29 3 Problem Description 32 3.1 Minimum Cost Network Flow Problem: Mincost Problem . 32 3.1.1 Mincost Problem 1: Allocation of indivisible resources . 34 3.1.2 Mincost Problem 2: Allocation of divisible resources . 36 3.2MaximumFlowProblem:MaxflowProblem.......... 37 3.3 Optimization with Multi-Modal Cost Function ......... 39 i 4 Approach and Simulation Results for Minimum Cost Network Flow Problem 42 4.1 Mincost Problem 1: Resource Allocation with Indivisible Re- sources............................... 42 4.2 Mincost Problem 2: Allocation of Divisible Resources ..... 45 4.3SimulationResults........................ 48 4.3.1 Mincost Problem 1: Resource Allocation with Invisible Resources......................... 48 4.3.2 Mincost Problem 2: Resource Allocation with Divisible Resources......................... 50 5 Approach and Simulation Results for Maximum Flow Prob- lem 57 5.1 Optimality Conditions ...................... 58 5.2DualityGap............................ 59 5.3 Distributed Primal and Dual Update .............. 60 5.4DualStepSize........................... 63 5.5PrimalStepSize......................... 64 5.6ConvergenceAnalysis....................... 68 5.6.1 Descent Direction ..................... 68 5.6.2 ConvergenceofPrimalVariables............. 71 5.6.3 ConvergenceofDualVariables.............. 73 5.7SimulationResults........................ 76 6 Approach and Simulation Results for Multi-Modal NUM Prob- lem 82 6.1SimulationResults........................ 86 ii 7 Applications 91 7.1OptimalPowerFlowProbleminPowerGrids......... 91 7.1.1 BuyerStrategy...................... 97 7.1.2 DealerStrategy...................... 99 7.2 Utility Maximization in Cloud Computing ........... 99 7.3SimulationResults........................ 104 7.3.1 OptimalPowerFlowProbleminPowerGrids..... 104 7.3.2 Cloud Computing Systems ................ 106 8 Dissertation Contributions and Future Scope 111 8.1 Dissertation Contributions .................... 111 8.2FutureScope........................... 115 iii List of Tables 4.1 Comparison between Marked Based Method and Integer Pro- gramming (Mincost Problem 1) ................. 52 4.2 Comparison between Marked Based Method and Linear Pro- gramming (Mincost Problem 2) ................. 52 4.3TaskDemand........................... 53 4.4ResourceCapacities........................ 53 5.1 Comparison Between the Proposed Distributed Optimization and Central Optimization
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