Convex Optimization

Convex Optimization

Convex Optimization Stephen Boyd Department of Electrical Engineering Stanford University Lieven Vandenberghe Electrical Engineering Department University of California, Los Angeles

cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S��ao Paolo, Delhi Cambridge University Press The Edinburgh Building, Cambridge, CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York http://www.cambridge.org Information on this title: www.cambridge.org/9780521833783 c circlecopyrt Cambridge University Press 2004 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2004 Seventh printing with corrections 2009 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library Library of Congress Cataloguing-in-Publication data Boyd, Stephen P. Convex Optimization / Stephen Boyd & Lieven Vandenberghe p. cm. Includes bibliographical references and index. ISBN 0 521 83378 7 1. Mathematical optimization. 2. Convex functions. I. Vandenberghe, Lieven. II. Title. QA402.5.B69 2004 519.6���dc22 2003063284 ISBN 978-0-521-83378-3 hardback Cambridge University Press has no responsiblity for the persistency or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

For Anna, Nicholas, and Nora Dani��el and Margriet

For Anna, Nicholas, and Nora Dani��el and Margriet

Contents Preface xi 1 Introduction 1 1.1 Mathematical optimization . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Least-squares and linear programming . . . . . . . . . . . . . . . . . . 4 1.3 Convex optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Nonlinear optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 I Theory 19 2 Convex sets 21 2.1 Affine and convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Some important examples . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Operations that preserve convexity . . . . . . . . . . . . . . . . . . . . 35 2.4 Generalized inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5 Separating and supporting hyperplanes . . . . . . . . . . . . . . . . . . 46 2.6 Dual cones and generalized inequalities . . . . . . . . . . . . . . . . . . 51 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3 Convex functions 67 3.1 Basic properties and examples . . . . . . . . . . . . . . . . . . . . . . 67 3.2 Operations that preserve convexity . . . . . . . . . . . . . . . . . . . . 79 3.3 The conjugate function . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.4 Quasiconvex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.5 Log-concave and log-convex functions . . . . . . . . . . . . . . . . . . 104 3.6 Convexity with respect to generalized inequalities . . . . . . . . . . . . 108 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

viii Contents 4 Convex optimization problems 127 4.1 Optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.2 Convex optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.3 Linear optimization problems . . . . . . . . . . . . . . . . . . . . . . . 146 4.4 Quadratic optimization problems . . . . . . . . . . . . . . . . . . . . . 152 4.5 Geometric programming . . . . . . . . . . . . . . . . . . . . . . . . . . 160 4.6 Generalized inequality constraints . . . . . . . . . . . . . . . . . . . . . 167 4.7 Vector optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 5 Duality 215 5.1 The Lagrange dual function . . . . . . . . . . . . . . . . . . . . . . . . 215 5.2 The Lagrange dual problem . . . . . . . . . . . . . . . . . . . . . . . . 223 5.3 Geometric interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 232 5.4 Saddle-point interpretation . . . . . . . . . . . . . . . . . . . . . . . . 237 5.5 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 5.6 Perturbation and sensitivity analysis . . . . . . . . . . . . . . . . . . . 249 5.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 5.8 Theorems of alternatives . . . . . . . . . . . . . . . . . . . . . . . . . 258 5.9 Generalized inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 264 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 II Applications 289 6 Approximation and fitting 291 6.1 Norm approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 6.2 Least-norm problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 6.3 Regularized approximation . . . . . . . . . . . . . . . . . . . . . . . . 305 6.4 Robust approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 6.5 Function fitting and interpolation . . . . . . . . . . . . . . . . . . . . . 324 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 7 Statistical estimation 351 7.1 Parametric distribution estimation . . . . . . . . . . . . . . . . . . . . 351 7.2 Nonparametric distribution estimation . . . . . . . . . . . . . . . . . . 359 7.3 Optimal detector design and hypothesis testing . . . . . . . . . . . . . 364 7.4 Chebyshev and Chernoff bounds . . . . . . . . . . . . . . . . . . . . . 374 7.5 Experiment design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

Contents ix 8 Geometric problems 397 8.1 Projection on a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 8.2 Distance between sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 8.3 Euclidean distance and angle problems . . . . . . . . . . . . . . . . . . 405 8.4 Extremal volume ellipsoids . . . . . . . . . . . . . . . . . . . . . . . . 410 8.5 Centering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 8.6 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 8.7 Placement and location . . . . . . . . . . . . . . . . . . . . . . . . . . 432 8.8 Floor planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 III Algorithms 455 9 Unconstrained minimization 457 9.1 Unconstrained minimization problems . . . . . . . . . . . . . . . . . . 457 9.2 Descent methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 9.3 Gradient descent method . . . . . . . . . . . . . . . . . . . . . . . . . 466 9.4 Steepest descent method . . . . . . . . . . . . . . . . . . . . . . . . . 475 9.5 Newton���s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 9.6 Self-concordance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 9.7 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 10 Equality constrained minimization 521 10.1 Equality constrained minimization problems . . . . . . . . . . . . . . . 521 10.2 Newton���s method with equality constraints . . . . . . . . . . . . . . . . 525 10.3 Infeasible start Newton method . . . . . . . . . . . . . . . . . . . . . . 531 10.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 11 Interior-point methods 561 11.1 Inequality constrained minimization problems . . . . . . . . . . . . . . 561 11.2 Logarithmic barrier function and central path . . . . . . . . . . . . . . 562 11.3 The barrier method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 11.4 Feasibility and phase I methods . . . . . . . . . . . . . . . . . . . . . . 579 11.5 Complexity analysis via self-concordance . . . . . . . . . . . . . . . . . 585 11.6 Problems with generalized inequalities . . . . . . . . . . . . . . . . . . 596 11.7 Primal-dual interior-point methods . . . . . . . . . . . . . . . . . . . . 609 11.8 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

x Contents Appendices 631 A Mathematical background 633 A.1 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 A.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 A.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 A.4 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 A.5 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 B Problems involving two quadratic functions 653 B.1 Single constraint quadratic optimization . . . . . . . . . . . . . . . . . 653 B.2 The S-procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 B.3 The field of values of two symmetric matrices . . . . . . . . . . . . . . 656 B.4 Proofs of the strong duality results . . . . . . . . . . . . . . . . . . . . 657 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 C Numerical linear algebra background 661 C.1 Matrix structure and algorithm complexity . . . . . . . . . . . . . . . . 661 C.2 Solving linear equations with factored matrices . . . . . . . . . . . . . . 664 C.3 LU, Cholesky, and LDLT factorization . . . . . . . . . . . . . . . . . . 668 C.4 Block elimination and Schur complements . . . . . . . . . . . . . . . . 672 C.5 Solving underdetermined linear equations . . . . . . . . . . . . . . . . . 681 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 References 685 Notation 697 Index 701

Preface This book is about convex optimization, a special class of mathematical optimiza- tion problems, which includes least-squares and linear programming problems. It is well known that least-squares and linear programming problems have a fairly complete theory, arise in a variety of applications, and can be solved numerically very efficiently. The basic point of this book is that the same can be said for the larger class of convex optimization problems. While the mathematics of convex optimization has been studied for about a century, several related recent developments have stimulated new interest in the topic. The first is the recognition that interior-point methods, developed in the 1980s to solve linear programming problems, can be used to solve convex optimiza- tion problems as well. These new methods allow us to solve certain new classes of convex optimization problems, such as semidefinite programs and second-order cone programs, almost as easily as linear programs. The second development is the discovery that convex optimization problems (beyond least-squares and linear programs) are more prevalent in practice than was previously thought. Since 1990 many applications have been discovered in areas such as automatic control systems, estimation and signal processing, com- munications and networks, electronic circuit design, data analysis and modeling, statistics, and finance. Convex optimization has also found wide application in com- binatorial optimization and global optimization, where it is used to find bounds on the optimal value, as well as approximate solutions. We believe that many other applications of convex optimization are still waiting to be discovered. There are great advantages to recognizing or formulating a problem as a convex optimization problem. The most basic advantage is that the problem can then be solved, very reliably and efficiently, using interior-point methods or other special methods for convex optimization. These solution methods are reliable enough to be embedded in a computer-aided design or analysis tool, or even a real-time reactive or automatic control system. There are also theoretical or conceptual advantages of formulating a problem as a convex optimization problem. The associated dual problem, for example, often has an interesting interpretation in terms of the original problem, and sometimes leads to an efficient or distributed method for solving it. We think that convex optimization is an important enough topic that everyone who uses computational mathematics should know at least a little bit about it. In our opinion, convex optimization is a natural next topic after advanced linear algebra (topics like least-squares, singular values), and linear programming.

xii Preface Goal of this book For many general purpose optimization methods, the typical approach is to just try out the method on the problem to be solved. The full benefits of convex optimization, in contrast, only come when the problem is known ahead of time to be convex. Of course, many optimization problems are not convex, and it can be difficult to recognize the ones that are, or to reformulate a problem so that it is convex. Our main goal is to help the reader develop a working knowledge of convex optimization, i.e., to develop the skills and background needed to recognize, formulate, and solve convex optimization problems. Developing a working knowledge of convex optimization can be mathematically demanding, especially for the reader interested primarily in applications. In our experience (mostly with graduate students in electrical engineering and computer science), the investment often pays off well, and sometimes very well. There are several books on linear programming, and general nonlinear pro- gramming, that focus on problem formulation, modeling, and applications. Several other books cover the theory of convex optimization, or interior-point methods and their complexity analysis. This book is meant to be something in between, a book on general convex optimization that focuses on problem formulation and modeling. We should also mention what this book is not. It is not a text primarily about convex analysis, or the mathematics of convex optimization several existing texts cover these topics well. Nor is the book a survey of algorithms for convex optimiza- tion. Instead we have chosen just a few good algorithms, and describe only simple, stylized versions of them (which, however, do work well in practice). We make no attempt to cover the most recent state of the art in interior-point (or other) meth- ods for solving convex problems. Our coverage of numerical implementation issues is also highly simplified, but we feel that it is adequate for the potential user to develop working implementations, and we do cover, in some detail, techniques for exploiting structure to improve the efficiency of the methods. We also do not cover, in more than a simplified way, the complexity theory of the algorithms we describe. We do, however, give an introduction to the important ideas of self-concordance and complexity analysis for interior-point methods. Audience This book is meant for the researcher, scientist, or engineer who uses mathemat- ical optimization, or more generally, computational mathematics. This includes, naturally, those working directly in optimization and operations research, and also many others who use optimization, in fields like computer science, economics, fi- nance, statistics, data mining, and many fields of science and engineering. Our primary focus is on the latter group, the potential users of convex optimization, and not the (less numerous) experts in the field of convex optimization. The only background required of the reader is a good knowledge of advanced calculus and linear algebra. If the reader has seen basic mathematical analysis (e.g., norms, convergence, elementary topology), and basic probability theory, he or she should be able to follow every argument and discussion in the book. We hope that

Preface xiii readers who have not seen analysis and probability, however, can still get all of the essential ideas and important points. Prior exposure to numerical computing or optimization is not needed, since we develop all of the needed material from these areas in the text or appendices. Using this book in courses We hope that this book will be useful as the primary or alternate textbook for several types of courses. Since 1995 we have been using drafts of this book for graduate courses on linear, nonlinear, and convex optimization (with engineering applications) at Stanford and UCLA. We are able to cover most of the material, though not in detail, in a one quarter graduate course. A one semester course allows for a more leisurely pace, more applications, more detailed treatment of theory, and perhaps a short student project. A two quarter sequence allows an expanded treatment of the more basic topics such as linear and quadratic programming (which are very useful for the applications oriented student), or a more substantial student project. This book can also be used as a reference or alternate text for a more traditional course on linear and nonlinear optimization, or a course on control systems (or other applications area), that includes some coverage of convex optimization. As the secondary text in a more theoretically oriented course on convex optimization, it can be used as a source of simple practical examples. Acknowledgments We have been developing the material for this book for almost a decade. Over the years we have benefited from feedback and suggestions from many people, including our own graduate students, students in our courses, and our colleagues at Stanford, UCLA, and elsewhere. Unfortunately, space limitations and shoddy record keeping do not allow us to name everyone who has contributed. However, we wish to particularly thank A. Aggarwal, V. Balakrishnan, A. Bernard, B. Bray, R. Cottle, A. d���Aspremont, J. Dahl, J. Dattorro, D. Donoho, J. Doyle, L. El Ghaoui, P. Glynn, M. Grant, A. Hansson, T. Hastie, A. Lewis, M. Lobo, Z.-Q. Luo, M. Mesbahi, W. Naylor, P. Parrilo, I. Pressman, R. Tibshirani, B. Van Roy, L. Xiao, and Y. Ye. J. Jalden and A. d���Aspremont contributed the time-frequency analysis example in ��6.5.4, and the consumer preference bounding example in ��6.5.5, respectively. P. Parrilo suggested exercises 4.4 and 4.56. Newer printings benefited greatly from Igal Sason���s meticulous reading of the book. We want to single out two others for special acknowledgment. Arkadi Ne- mirovski incited our original interest in convex optimization, and encouraged us to write this book. We also want to thank Kishan Baheti for playing a critical role in the development of this book. In 1994 he encouraged us to apply for a Na- tional Science Foundation combined research and curriculum development grant, on convex optimization with engineering applications, and this book is a direct (if delayed) consequence. Stephen Boyd Stanford, California Lieven Vandenberghe Los Angeles, California

Preface xiii readers who have not seen analysis and probability, however, can still get all of the essential ideas and important points. Prior exposure to numerical computing or optimization is not needed, since we develop all of the needed material from these areas in the text or appendices. Using this book in courses We hope that this book will be useful as the primary or alternate textbook for several types of courses. Since 1995 we have been using drafts of this book for graduate courses on linear, nonlinear, and convex optimization (with engineering applications) at Stanford and UCLA. We are able to cover most of the material, though not in detail, in a one quarter graduate course. A one semester course allows for a more leisurely pace, more applications, more detailed treatment of theory, and perhaps a short student project. A two quarter sequence allows an expanded treatment of the more basic topics such as linear and quadratic programming (which are very useful for the applications oriented student), or a more substantial student project. This book can also be used as a reference or alternate text for a more traditional course on linear and nonlinear optimization, or a course on control systems (or other applications area), that includes some coverage of convex optimization. As the secondary text in a more theoretically oriented course on convex optimization, it can be used as a source of simple practical examples. Acknowledgments We have been developing the material for this book for almost a decade. Over the years we have benefited from feedback and suggestions from many people, including our own graduate students, students in our courses, and our colleagues at Stanford, UCLA, and elsewhere. Unfortunately, space limitations and shoddy record keeping do not allow us to name everyone who has contributed. However, we wish to particularly thank A. Aggarwal, V. Balakrishnan, A. Bernard, B. Bray, R. Cottle, A. d���Aspremont, J. Dahl, J. Dattorro, D. Donoho, J. Doyle, L. El Ghaoui, P. Glynn, M. Grant, A. Hansson, T. Hastie, A. Lewis, M. Lobo, Z.-Q. Luo, M. Mesbahi, W. Naylor, P. Parrilo, I. Pressman, R. Tibshirani, B. Van Roy, L. Xiao, and Y. Ye. J. Jalden and A. d���Aspremont contributed the time-frequency analysis example in ��6.5.4, and the consumer preference bounding example in ��6.5.5, respectively. P. Parrilo suggested exercises 4.4 and 4.56. Newer printings benefited greatly from Igal Sason���s meticulous reading of the book. We want to single out two others for special acknowledgment. Arkadi Ne- mirovski incited our original interest in convex optimization, and encouraged us to write this book. We also want to thank Kishan Baheti for playing a critical role in the development of this book. In 1994 he encouraged us to apply for a Na- tional Science Foundation combined research and curriculum development grant, on convex optimization with engineering applications, and this book is a direct (if delayed) consequence. Stephen Boyd Stanford, California Lieven Vandenberghe Los Angeles, California

Chapter 1 Introduction In this introduction we give an overview of mathematical optimization, focusing on the special role of convex optimization. The concepts introduced informally here will be covered in later chapters, with more care and technical detail. 1.1 Mathematical optimization A mathematical optimization problem, or just optimization problem, has the form minimize f0(x) subject to fi(x) ��� bi, i = 1,...,m. (1.1) Here the vector x = (x1,...,xn) is the optimization variable of the problem, the function f0 : Rn ��� R is the objective function, the functions fi : Rn ��� R, i = 1,...,m, are the (inequality) constraint functions, and the constants b1,...,bm are the limits, or bounds, for the constraints. A vector x��� is called optimal, or a solution of the problem (1.1), if it has the smallest objective value among all vectors that satisfy the constraints: for any z with f1(z) ��� b1,...,fm(z) ��� bm, we have f0(z) ��� f0(x���). We generally consider families or classes of optimization problems, characterized by particular forms of the objective and constraint functions. As an important example, the optimization problem (1.1) is called a linear program if the objective and constraint functions f0,...,fm are linear, i.e., satisfy fi(��x + ��y) = ��fi(x) + ��fi(y) (1.2) for all x, y ��� Rn and all ��, �� ��� R. If the optimization problem is not linear, it is called a nonlinear program. This book is about a class of optimization problems called convex optimiza- tion problems. A convex optimization problem is one in which the objective and constraint functions are convex, which means they satisfy the inequality fi(��x + ��y) ��� ��fi(x) + ��fi(y) (1.3)