Inexact interior point methods for large scale linear and convex quadratic semidefinite programming

INEXACT INTERIOR-POINT METHODS FOR LARGE SCALE LINEAR AND CONVEX QUADRATIC SEMIDEFINITE PROGRAMMING LI LU (B.Sc., SJTU, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2010 To my parents Acknowledgements I would like to express my heartfelt gratitude to my advisor Professor Toh KimChuan, for his invaluable guidance and expertise in optimization, his utmost support and encouragement throughout the past five years Without him, this thesis would never have been possible The way of conducting scientific research, the opening posture towards new ideas and the attitude on teaching that I learned from him would be a lifelong treasure I would like to express my sincere thanks to Professor Zhao Gongyun for his instruction on game theory and numerical optimization, which are the first and the last modules I took during my study in NUS I sincerely thank him for sharing with me his wisdom and experience in the field of numerical computation and optimization theory I am also indebted to Professor Sun Defeng for his continuous effort on conducting weekly optimization seminars in Department of Mathematics, NUS His broad knowledge and enthusiasm on optimization have helped me tremendously in exploring various topics I am also thankful to Dr Liu Yongjin, Dr Yun Sangwoon and Dr Zhao Xinyuan v vi Acknowledgements for their helpful and constructive discussions in many topics related to my thesis This acknowledgement will remain incomplete without expressing my gratitude to my fellow colleagues and friends at NUS, where many happy memories I will carry from My thanks also goes out to National University of Singapore and Department of Mathematics for providing me excellent working conditions and scholarships to complete my study Contents Acknowledgements Summary v xi List of Tables xiii Notation xvi Introduction 1.1 The bottleneck of interior-point methods 1.2 Organization of the thesis 1.3 Convex quadratic SDP 1.4 Sparse covariance selection 10 1.5 Dual-scaling interior-point methods 16 Symmetric cones and Euclidean Jordan Algebras 19 vii viii Contents Polynomial-time inexact interior-point methods for convex quadratic programming over symmetric cones 29 3.1 Convex quadratic symmetric cone programming 30 3.2 An infeasible central path and its neighborhood 33 3.3 An inexact infeasible interior-point algorithm 38 3.4 Proof of Lemma 3.7 44 Inexact primal-dual path-following methods for l1 -regularized logdeterminant semidefinite programming problem 57 4.1 A customized inexact primal-dual interior-point method 58 4.2 Preconditioners 64 4.3 Computation of search direction for the special case (1.8) 66 4.3.1 Computing (∆x, ∆y) first 67 4.3.2 Computing (∆y, ∆u) first 69 Numerical Experiment 71 4.4.1 Synthetic Examples 72 4.4.2 Real world examples 78 4.4 An inexact dual-scaling interior-point method for linear programming problems over symmetric cones 83 5.1 An inexact dual-scaling interior point algorithm 84 5.1.1 Inexact search directions 87 5.2 Verification of the admissible condition (5.13b) 95 5.3 A practical inexact-direction dual-scaling algorithm 98 5.4 Numerical experiments 103 Conclusions and future work 111 Contents Bibliography ix 113 114 Bibliography [6] J A Bilmes, Natural statistical models for automatic speech recognition, PhD thesis, University of California, Berkeley, 1999 [7] P Biswas, K.-C Toh, and Y Ye, A distributed SDP approach for largescale noisy anchor-free graph reailzation with applications to molecular conformation, SIAM J Sci Comput., 30 (2008), pp 1251–1277 [8] B Borchers, SDPLIB 1.2, library of semidefinite programming test problems, Optim Methods Softw., 11/12 (1999), pp 683–690 Interior point methods [9] R Borsdorf and N J Higham, A preconditioned Newton algorithm for the nearest correlation matrix, IMA J Numer Anal., 30 (2010), pp 94–107 [10] S Boyd and L Xiao, Least-squares covariance matrix adjustment, SIAM J Matrix Anal Appl., 27 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Therefore the interior- point methods using iterative solvers are commonly called inexact interior- point methods In order to guarantee the global polynomial convergence of inexact interior- point methods,