Convex Optimization-凸优化.pdf

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Convex Optimization-凸优化.pdf Convex Optimization-凸优化.pdf Convex Optimization-凸优化.pdf
Convex Optimization Stephen Boyd Department of elcctrical enginccring Stanford university Lieven Vandenberghe Electrical Engineering Department University of California, Los Angeles 圖 CAMBRIDGE 圆 UNIVERSITY PRESS AMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paolo, Delhi Cambridge University Press The Edinburgh Building, Cambridge, CB2 &RU, UK Published in the United States of America by Cambridge University Press, New York http://www.cambridge.org Informationonthistitlewww.cambridge.org/9780521833783 (c Cambridge University Press 2004 This publicalion 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 Includes bibliographical references and index ISBN0521833787 1. Mathematical optimization. 2. Convex functions. I. Vandenberghe, Lieven. II. Title QA402.5.B692004 519.6dc222003063281 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 e s referre d to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate Anna Nicholas, and nora Daniel and Margriet Contents Preface 1 Introduction 1.1 Mathematical optimizati 1.2 Least-squares and linear programming 1.3 Convex optimization 1.4 Nonlinear optimization 1.5 Outli 11 1. 6 Notation 14 Bibliograph 16 Theo 19 2 Convex sets 21 2.1 Affine and convex sets 21 2.2 Some important examples 2 2.3 Operations that preserve convexity 35 2.4 Generalized inequalities 2.5 Separating and supporting hyperplanes 2.6 Dual cones and generalized inequalities 51 Bibliography 59 60 3 Convex functions 67 3. 1 Basic properties and 3.2 Operations that preserve convexity 79 3.3 The col 90 3.4 Quasiconvex functions 95 3.5 Log-concave and log 104 3.6 Convexity with respect to generalized inequalities Bibliograph xercises Content 4 Convex optimization problems 127 4.1 Optimization problems 127 4.2 Convex optimization 136 4.3 Linear optimization problem 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 Xercises ..189 5 Duali 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 anal ysis 249 5.7 Examples 253 5.8 Theorems of alternatives ..258 5.9 Generalized inequalities 264 Bibliography 272 Exercises 273 l1 Applications 289 6 Approximation and fitting 291 6.1 Norm approximation 291 6.2 Least-norm problems 302 3 Regularized approxin 305 6.4 Robust approximation 318 6.5 Function fitting and interpolation 324 Bibliography 343 xercises 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 bound 374 7.5 Experiment design 384 Bibliograph 392 Exercises ..393 Contents 8 Geometric problems 397 8.1 Projection on a set 397 8.2 Distance between sets 402 8. 3 Euclidean distance and angle problems 8.4 Extremal volume ellipsoids 8.5 Centering 416 8.6 Classification 422 8.7 Placement and location .432 8.8 Floor planning 438 Bibliography 446 Exercises I Al gorithms 455 9 Unconstrained minimization 457 9.1 Unconstrained minimization problems 457 9.2 Descent methods 463 9.3 Gradient descent method 466 9. 4 Ste descent method 9.5 Newton's method 484 9.6 Self- 9.7 Implementation 508 graphy 514 10 Equality constrained 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 Bibliography 556 E× excises 557 11 Interior-point methods 561 l1. 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 11.5 Complexity analysis via self-concordance 11.6 Problems with generalized inequalities 596 11.7 Primal-dual interior-point methods 609 11. 8 Implementation 615 Bibliography 621 Exercises 623 Content appendices 631 A Mathematical background 633 A1 Norm 633 A2 Anal 637 A 3 Functions 639 A 4D 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 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 C2 Solving linear equations with factored matrices 664 C 3 LU, Cholesky, and LDl factorization ...,..668 C 4 block elimination and schur complements ..672 C 5 Solving underdetermined linear equations 681 Bibliography 684 References 685 notatio 697 Index 701

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