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Distributed Optimization Algorithms for Networked Systems Distributed Optimization Algorithms for Networked Systems by Nikolaos Chatzipanagiotis Department of Mechanical Engineering & Materials Science Duke University Date: Approved: Michael M. Zavlanos, Supervisor Wilkins Aquino Henri Gavin Alejandro Ribeiro Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mechanical Engineering & Materials Science in the Graduate School of Duke University 2015 Abstract Distributed Optimization Algorithms for Networked Systems by Nikolaos Chatzipanagiotis Department of Mechanical Engineering & Materials Science Duke University Date: Approved: Michael M. Zavlanos, Supervisor Wilkins Aquino Henri Gavin Alejandro Ribeiro An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mechanical Engineering & Materials Science in the Graduate School of Duke University 2015 Copyright c 2015 by Nikolaos Chatzipanagiotis All rights reserved except the rights granted by the Creative Commons Attribution-Noncommercial Licence Abstract Distributed optimization methods allow us to decompose certain optimization prob- lems into smaller, more manageable subproblems that are solved iteratively and in parallel. For this reason, they are widely used to solve large-scale problems arising in areas as diverse as wireless communications, optimal control, machine learning, artificial intelligence, computational biology, finance and statistics, to name a few. Moreover, distributed algorithms avoid the cost and fragility associated with cen- tralized coordination, and provide better privacy for the autonomous decision mak- ers. These are desirable properties, especially in applications involving networked robotics, communication or sensor networks, and power distribution systems. In this thesis we propose the Accelerated Distributed Augmented Lagrangians (ADAL) algorithm, a novel decomposition method for convex optimization prob- lems. The method is based on the augmented Lagrangian framework and addresses problems that involve multiple agents optimizing a separable convex objective func- tion subject to convex local constraints and linear coupling constraints. We establish the convergence of ADAL and also show that it has a worst-case Op1{kq convergence rate, where k denotes the number of iterations. Moreover, we show that ADAL converges to a local minimum of the problem for cases with non-convex objective functions. This is the first published work that formally establishes the convergence of a distributed augmented Lagrangian method for non-convex optimization problems. An alternative way to select the stepsizes iv used in the algorithm is also discussed. Furthermore, we consider cases where the distributed algorithm needs to operate in the presence of uncertainty and noise and show that the generated sequences of primal and dual variables converge to their respective optimal sets almost surely. In particular, we are concerned with scenarios where: i) the local computation steps are inexact or are performed in the presence of uncertainty, and ii) the message exchanges between agents are corrupted by noise. In this case, the proposed scheme can be classified as a distributed stochastic approximation method. Compared to existing literature in this area, our work is the first that utilizes the augmented Lagrangian framework. Moreover, the method allows us to solve a richer class of problems as compared to existing methods on distributed stochastic approximation that consider only consensus constraints. Extensive numerical experiments have been carried out in an effort to validate the novelty and effectiveness of the proposed method in all the areas of the afore- mentioned theoretical contributions. We examine problems in convex, non-convex, and stochastic settings where uncertainties and noise affect the execution of the al- gorithm. For the convex cases, we present applications of ADAL to certain popular network optimization problems, as well as to a two-stage stochastic optimization problem. The simulation results suggest that the proposed method outperforms the state-of-the-art distributed augmented Lagrangian methods that are known in the literature. For the non-convex cases, we perform simulations on certain simple non-convex problems to establish that ADAL indeed converges to non-trivial local solutions of the problems; in comparison, the straightforward implementation of the other distributed augmented Lagrangian methods on the same problems does not lead to convergence. For the stochastic setting, we present simulation results of ADAL applied on network optimization problems and examine the effect that noise and uncertainties have in the convergence behavior of the method. v As an extended and more involved application, we also consider the problem of relay cooperative beamforming in wireless communications systems. Specifically, we study the scenario of a multi-cluster network, in which each cluster contains multiple single-antenna source destination pairs that communicate simultaneously over the same channel. The communications are supported by cooperating amplify- and-forward relays, which perform beamforming. Since the emerging problem is non- convex, we propose an approximate convex reformulation. Based on ADAL, we also discuss two different ways to obtain a distributed solution that allows for autonomous computation of the optimal beamforming decisions by each cluster, while taking into account intra- and inter-cluster interference effects. Our goal in this thesis is to advance the state-of-the-art in distributed optimiza- tion by proposing methods that combine fast convergence, wide applicability, ease of implementation, low computational complexity, and are robust with respect to delays, uncertainty in the problem parameters, noise corruption in the message ex- changes, and inexact computations. vi `To my parents and friends' vii Contents Abstract iv List of Figures xi Acknowledgements xvii 1 Introduction1 1.1 Motivation.................................1 1.1.1 Distributed Optimization.....................1 1.1.2 Stochastic Settings........................4 1.2 Relevant Literature............................5 1.2.1 Distributed Optimization.....................5 1.2.2 Stochastic Settings........................7 1.3 Thesis contents..............................8 1.4 Contributions............................... 11 2 Preliminaries 14 2.1 Convex Analysis.............................. 15 2.2 Optimality Conditions.......................... 18 2.2.1 Unconstrained Optimization................... 18 2.2.2 Constrained Optimization.................... 19 2.3 Duality Theory.............................. 22 2.3.1 Lagrangian Duality........................ 22 viii 2.3.2 Dual Subgradient Method.................... 26 2.3.3 Dual Decomposition....................... 28 2.4 Augmented Lagrangians......................... 30 2.4.1 The augmented Lagrangian Method............... 31 2.4.2 The Alternating Directions Method of Multipliers....... 34 2.4.3 The Diagonal Quadratic Approximation Method........ 36 3 The Accelerated Distributed Augmented Lagrangians Method 39 3.1 The ADAL Algorithm.......................... 40 3.2 Convergence................................ 43 3.3 Rate of Convergence........................... 55 3.4 Convergence for Non-convex Problems and Alternative Stepsize Choices 64 3.4.1 Alternative Stepsize Choices................... 65 3.4.2 Convergence for Non-convex Problems............. 67 3.5 The Stochastic ADAL.......................... 85 3.5.1 The ADAL in a Stochastic Setting............... 87 3.5.2 Almost Sure Convergence of Stochastic ADAL......... 92 4 Numerical Analysis 112 4.1 Network Utility Maximization Problems................ 112 4.2 Network Flow Problems......................... 118 4.3 Two-Stage Stochastic Capacity Expansion problems.......... 125 4.3.1 Two-Stage Stochastic Optimization Problems......... 128 4.3.2 The Two-Stage Capacity Expansion Problem......... 130 4.3.3 Capacity Expansion Decomposition with respect to the Net- work Nodes............................ 132 4.3.4 Distributed Solution using the ADAL method......... 134 4.3.5 Numerical Analysis........................ 138 ix 4.4 Non-convex Problems........................... 142 4.5 Problems with Noise and Uncertainties................. 149 5 Application to Relay Beamforming for Wireless Communications 156 5.1 Relay Beamforming............................ 159 5.1.1 Single Cluster case........................ 159 5.1.2 Multiple Clusters case...................... 165 5.2 Distributed Relay Beamforming..................... 172 5.2.1 Direct method........................... 173 5.2.2 Indirect Method.......................... 177 5.3 Numerical Analysis............................ 181 5.3.1 Discussion............................. 186 6 Conclusions and Future Work 188 6.1 Summary of Main Results........................ 188 6.2 Future Directions............................. 190 6.2.1 Asynchronous Implementation.................. 191 6.2.2 More Advanced Stochastic Formulations............ 192 Bibliography 194 Biography 212 x List of Figures 4.1 Random network of 50 sources (red dots) and 2 sinks (green squares). The rate of flow tij through arc pi; jq defines the thickness and color of the corresponding drawn line. Thin, light blue lines indicate weaker links..................................... 114 4.2 Evolution of the sum of rates iPS si during implementation of ADAL. The horizontal line depicts
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