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Rank-Constrained Optimization: Algorithms and Applications Rank-Constrained Optimization: Algorithms and Applications Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Chuangchuang Sun, BS Graduate Program in Aeronautical and Astronautical Engineering The Ohio State University 2018 Dissertation Committee: Ran Dai, Advisor Mrinal Kumar Manoj Srinivasan Wei Zhang c Copyright by Chuangchuang Sun 2018 Abstract Matrix rank related optimization problems comprise rank-constrained optimization problems (RCOPs) and rank minimization problems, which involve the matrix rank function in the objective or the constraints. General RCOPs are defined to optimize a convex objective function subject to a set of convex constraints and rank constraints on unknown rectangular matrices. RCOPs have received extensive attention due to their wide applications in signal processing, model reduction, and system identification, just to name a few. Noteworthily, The general/nonconvex quadratically constrained quadratic programming (QCQP) problem with wide applications is a special category of RCOP. On the other hand, when the rank function appears in the objective of an optimization problem, it turns out to be a rank minimization problem (RMP), classified as another category of nonconvex optimization. Applications of RMPs have been found in a variety of areas, such as matrix completion, control system analysis and design, and machine learning. At first, RMP and RCOPs are not isolated. Though we can transform a RMP into a RCOP by introducing a slack variable, the reformulated RCOP, however, has an unknown upper bound for the rank function. That is a big shortcoming. Provided that a given matrix upper bounded is a prerequisite for most of the algorithms when solving RCOPs, we first fill this gap by demonstrating that RMPs can be equivalently transformed into RCOPs by introducing a quadratic matrix equality constraint. Furthermore, the wide application of RMPs and RCOPs attract extensive studies aiming at developing efficient optimization ii algorithms to solve the challenging matrix-related optimization problems. Motivated by the limitations of existing algorithms, we aim to develop effective and efficient algorithms to solve rank-related optimization problems. Two classes of algorithms and their variants are proposed here. The first algorithm, based on the alternating minimization, is named iterative rank mini- mization (IRM). The IRM method, with each sequential problem formulated as a convex optimization problem, aims to solve general RCOPs, where the constrained rank could be any assigned integer number. Although IRM is primarily designed for RCOPs with rank constraints on positive semidefinite matrices, a semidefinite embedding lemma is introduced to extend IRM to RCOPs with rank constraints on general rectangular matrices. Moreover, The proposed IRM method is applicable to RCOPs with rank inequalities constrained by upper or lower bounds, as well as rank equality constraints. Convergence of IRM is proved via the duality theory and the Karush-KuhnTucker conditions. Furthermore, the conversion from RMPs to RCOPs and the proposed IRM method are applied to solve cardinality mini- mization problems and cardinality-constrained optimization problems, which are handled as special cases of RMPs and RCOPs, respectively. To verify the effectiveness and improved performance of the proposed IRM method, representative applications, including matrix completion, system identification, output feedback stabilization, structured H2 controller design, and cardinality minimization problems, UAV path planning, network identification, are presented with comparative results. Moreover, for sparse QCQPs, a variant of IRM is developed to enhance the efficiency by exploiting the sparsity. The second algorithm, based on the framework of alternating direction method of mul- tipliers (ADMM), aims to get simple subproblem solutions, even in closed forms. We reformulate the rank constraint by matrix factorization and thus lead to the Burer-Monteiro iii form to achieve simpler subproblems. Despite the nonconvex constraint, we prove the ADMM iterations converge to a stationary point in two types of formulations under mild assumptions. Additionally, recent work suggests that in this latter form, when the matrix factors are wide enough, local optimum with high probability is also the global optimum. To demonstrate the scalability of our algorithm, we include results for MAX-CUT, community detection, and image segmentation benchmark and simulated examples. A variant of our algorithm reformulates the original RCOP using an approximate representation. The approx- imate formulation and the introduced bilinear constraint make each subproblem of ADMM more computationally-efficient. In particular cases, the subproblems can even be solved in a closed form. We then prove the global convergence of our proposed algorithm to a local minimizer of the approximate problem. Another variant focuses on solving QCQPs with both equality and inequality constraints. Based on inexact augmented Lagrangian method, the subproblem also admits a simple closed-form solution. By exploiting the structure of specific problems, such as network module detection, the computational efficiency can be further improved. Moreover, we also examine the simultaneous design of the network topology and the corresponding edge weights in presence of a cardinality constraint on the edge set. Network properties of interest for this design problem lead to optimization formulations with convex objectives, convex constraint sets, and cardinality constraints. This class of problems is referred to as cardinality-constrained optimization problem (CCOP); the cardinality constraint generally makes CCOPs NP-hard. In this work, a customized ADMM algorithm aiming to improve the scalability of the solution strategy for large-scale CCOPs is proposed. This algorithm utilizes the special structure of the problem formulation to obtain closed-form solutions for each iterative step of the corresponding ADMM updates. We also provide the iv convergence proof of the proposed customized ADMM to a stationary point under certain conditions. In conclusion, this dissertation developed the algorithmic frameworks to solve RCOPs and RMPs with convergence analysis. Comparative simulation results with the state-of-art algorithms are provided to validate the efficacy and effectiveness of proposed algorithms. v Dedicated to my parents vi Acknowledgments I would like to appreciate whoever helped me in this incredible journey. First and foremost, I would like to thank my advisor Dr. Ran Dai, for her support, encouragement throughout all this PhD program. Her technical and editorial suggestions are essential to this dissertation and I learnt a lot from her on academic research. Also, my thanks goes to my committee members, Dr. Mrinal Kumar, Dr. Manoj Srinivasan and Dr. Wei Zhang as well as my previous committee members back at Iowa State. Their insightful questions greatly help me to improve the quality of this dissertation. Moreover, I owe a thanks to the administrative staffs both at Ohio State and Iowa State, who have always been helpful in the administrative processes. The friendship with the colleagues in our lab is also much appreciated. It has been a pleasure to work with Yue Zu, Changhuang Wan, Nathaniel Kingry and Yen-chen Liu. The meaningful discussions also contribute to this dissertation. Moreover, I also appreciate the friendship with the kind people during my stay at both Ames, IA and Columbus, OH. That has been a great memory. Last but not the least, my deepest gratitude goes to my family, especially my parents Shikui Sun and Yufang Liu. Their unconditional love and support not only in this PhD program but ever since my birth, shape me the one who I am today. Surely this dissertation will not be possible without their support. Also, I wold like to thank my grandparents, my aunt, my sister and my brother, for everything. vii Vita August 2018 . PhD, Aeronautical and Astronautical Engineer- ing, The Ohio State University, USA. June 2013 . Bachelor of Science, Beihang University. Publications Research Publications Sun, Chuangchuang, and Ran Dai. "Topology Design and Identification for Dynamic Networks." Cooperative Control of Multi-Agent Systems: Theory and Applications (2017): 209. Sun, Chuangchuang, Ran Dai and Mehran Mesbahi. "Weighted Network Design with Cardinality Constraints via Alternating Direction Method of Multipliers." IEEE Transactions on Control of Network Systems, 2018, DOI: 10.1109/TCNS.2018.2789726 Sun, Chuangchuang, and Ran Dai. "Rank-constrained optimization and its applications." Automatica 82 (2017): 128-136. Sun, Chuangchuang, Yen-chen Liu and Ran Dai. "Two Approaches for Path Planning of Unmanned Aerial Vehicles with Avoidance Zones." Journal of Guidance, Control, and Dynamics (2017) 1:8. Ran Dai and Chuangchuang Sun, “Path Planning of Spatial Rigid Motion with Constrained Attitude”, Journal of Guidance, Control, and Dynamics, Vol. 38, No. 8, 2015, pp. 1356- 1365. viii Wang, Dangxiao, Jing Guo, Chuangchuang Sun, Mu Xu, and Yuru Zhang. "A Flexible Concept for Designing Multiaxis Force/Torque Sensors Using Force Closure Theorem." Instrumentation and Measurement, IEEE Transactions on62, no. 7 (2013): 1951-1959. Du, Jianbin, and Chuangchuang Sun. "Reliability-based vibro-acoustic microstructural topology optimization." Structural and Multidisciplinary Optimization 55.4 (2017): 1195- 1215 Wang, Xiaohui, Changhuang
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