INTRODUCTION TO NONLINEAR OPTIMIZATION

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INTRODUCTION TO NONLINEAR OPTIMIZATION Theory, Algorithms, and Applications with MATLAB. 讲解非线性优化方法的书籍。
MOS-SIAM Series on Optimization This series is published jointly by the mathematical Optimization Society and the Society for Industrial and Applied Mathematics. It includes research monographs, books on applications, textbooks at all levels, and tutorials. Besides being of high scientific quality, books in the series must advance the understanding and practice of optimization. They must also be written clearly and at an appropriate level for the intended audience Editor-in-Chief Katya Scheinberg Lehigh University Editorial Board Santanu S Dey, Georgia Institute of Technology Maryam Fazel, University of Washington Andrea Lodi, University of bologna Arkadi Nemirovski, Georgia Institute of Technology Stefan Ulbrich, Technische Universitat Darmstadt Luis Nunes Vicente, University of Coimbra David Williamson, Cornell University Stephen J. Wright, University of Wisconsin Series volumes Beck, Amir, Introduction to Nonlinear Optimization Theory, algorithms, and Applications with MATLAB Attouch, Hedy Buttazzo, Giuseppe, and michalle, gerard, Variational Analysis in Sobolev and BV Spaces Applications to PDEs and Optimization, Second edition Shapiro, Alexander, Dentcheva, Darinka, and Ruszczynski, Andrzej, Lectures on Stochastic Programming Modeling and Theory, Second Edition Locatell, Marco and Schoen, Fabio, Global Optimization: Theory, Algorithms, and Applications De Loera, Jesus a, Hemmecke, Raymond, and Koppe, Matthias, Algebraic and Geometric ldeas in the Theory of Discrete Optimization Blekherman, Grigor]y, Parrilo, Pablo A, and Thomas, Rekha r, editors, Semidefinite Optimization and Convex Algebraic Geometry Delfour, M. C, Introduction to Optimization and Semidifferential Calculus Ulbrich, Michael, Semismooth Newton Methods for Variational inequalities and Constrained Optimization Problems in Function Spaces Biegler, Lorenz T, Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes Shapiro, Alexander, Dentcheva, Darinka, and Ruszczynski, Andrzej, Lectures on Stochastic Programming Modeling and Theory Conn, Andrew R, Scheinberg, Katya, and vicente, Luis N, Introduction to Derivative-Free Optimization Ferris, Michael C, Mangasarian, Olvi L, and Wright, Stephen J, Linear Programming with MATLAB Attouch, Hedy, Buttazzo, Giuseppe, and Michalle, Gerard, Variational Analysis in Sobolev and BV Spaces Applications to PDEs and Optimization Wallace, Stein w. and Ziemba, William T, editors, Applications of Stochastic Programming Grotschel, Martin, editor, The Sharpest Cut: The impact of Manfred Padberg and His Work Renegar, James, A Mathematical View of Interior-Point Methods in Convex Optimization Ben-Tal, Aharon and Nemirovski, Arkadi, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications Conn, Andrew R, Gould, Nicholas L. M, and Toint, Phillippe L, Trust-Region Methods INTRODUCTION TO NONLINEAR OPTIMIZATION Theory, Algorithms, and Applications with MATLAB Amir Beck Technion-Israel Institute of Technology Kfar Saba. Israel SUIL Mathematical Optimization Society Society for Industrial and Applied Mathematics Mathematical Optimization Society Philadelphi Philadelphia Copyright o 2014 by the Society for Industrial and Applied Mathematics and the mathematical Optimization Society 10987654321 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA Trademarked names may be used in this book without the inclusion of a trademark symbol These names are used in an editorial context only; no infringement of trademark is intended MATLAB is a registered trademark of The MathWorks, Inc For MATLAB product information please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA01760-2098 USA 508-647-7000.Fax:508-647-7001.info@mathworks.com,www.mathworks.com Library of Congress Cataloging-in-Publication Data Beck Amir. author Introduction to nonlinear optimization theory, algorithms, and applications with MATLAB Amir Beck, Technion-Israel Institute of Technology kfar Saba, Israel pages cm-(MOS SIAM series on optimization Includes bibliographical references and index SBN978-1-611973-64-8 1. Mathematical optimization. 2. Nonlinear theories. 3. MATLAB. I. Title QA402.5.B42242014 519.6dc23 2014029493 si is a registered trademark Mathematical Optimization Society is a registered trademark ○f My wife Nili My daughters noy and Vered My parents Nili and Itzhak Contents Preface 1 Mathematical Preliminaries The S le Space R 鲁音 1.2 The space rmxn 1.3 Inner Products and norms 1.4 Eigenvalues and Eigenvectors 12256 1.5 Basic Topological Concepts....,,,... E Excises 10 2 Optimality Conditions for Unconstrained Optimization 2.1 Global and Local optima 13 2.2 Classification of matrices 2.3 Second Order Optimality conditions 2.4 Global Optimality Conditions 鲁··鲁◆ 30 2.5 Quadratic Functions 32 E eXercises 34 3 Least Squares 37 3.1“ Solution” of Overdetermined Systems. 37 3.2 Data Fittin 39 3.3 Regularized Least Squa 41 3.4 noising 8 42 3. li onlinear least squ ares 45 3.6 Circle Fitting 45 E 47 4 The Gradient method 49 4.1 Descent Directions metho 49 4.2 The Gradient Method 52 4. T C he conditio on Number 58 4.4 Diagonal Scaling 63 4.5 The Gauss-Newton Method 67 4.6 The Fermat-Weber problem 68 4.7 Convergence Analysis of the gradient Method 73 E 79 5 Newton' s Method 83 5.1 Pure Newton's Method 83 Contents 5.2 Damped Newton's Method 88 3 The Cholesky Factorization 90 Excises 94 6 Convex sets 6. 1 Definition and Example 97 2 Algebraic Operations with Convex Sets ..100 6.3 The Convex hull 101 6.4 Convex Cones 104 65 Topological Properties of Convex Sets.………….108 6.6 Extreme points 111 Exercises 113 7 Convex Functions 117 71 Definition and Examples.……………….117 7.2 First Order Characterizations of convex functions ,119 7. 3 Second Order Characterization of Convex Functions 123 74 Operations Preserving Convexity ,125 7.5 Level Sets of convex functions ...,.,....,. 130 7.6 Continuity and Differentiability of Convex Functions 132 7.7 Extended real- valued fl 135 7. 8 Maxima of convex Functions .,,,,,,,,,,.137 7.9 Convexity and inequalities 139 E exercises 8 Convex Optimization 147 8,1 Definition,,,,,,,,,,,,,,,,,,,,,,,,, 147 8.2 Exampl 8.3 The Orthogonal Projection Operator 156 84CVⅩ,,,,,, 158 Exercises 166 9 Optimization over a Convex Set 169 9.1 Stationarity .,,,,。。,,,,,,,,,.,,,,.169 9.2 Stationarity in Convex Problems ,173 9.3 The Orthogonal Projection Revisited 173 9.4 The gradient Projection Method ...,.175 9.5 Sparsity Constrained Problems ............ 183 Exercises ..,,,189 10 Optimality Conditions for Linearly Constrained Problems 191 10.1 Separation and Alternative Theorems ,,,,,,191 10.2 The KKt conditions 195 10.3 Orthogonal regression 203 E Excises 205 11 The KKt Conditions 207 11.1 Inequality Constrained Problems ,,,,,,,,,,,,207 11.2 Inequality and equality constrained problems 210 11. 3 The Convex Case 213 11.4 Constrained Least Squares 218 Contents 11.5 Second Order Optimality Conditions ,222 11.6 Optimality Conditions for the Trust region Subproblem.... 227 11.7 Total Least Squares 230 Exercises 233 12 Dualit 237 12. 1 Motivation and definition 237 12.2 Strong duality in the Convex Case ,,241 12. 3 Example p 247 270 Bibliographic No 275 Bibliography 277 nex 81

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