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MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® soft- ware or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2015 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20150205 International Standard Book Number-13: 978-1-4987-2115-8 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit- ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com © 2015 by Taylor & Francis Group, LLC

Contents Preface ......................................................................................................................xi Author .....................................................................................................................xv 1. Introduction .....................................................................................................1 1.1 Historical Review ..................................................................................1 1.2 Optimization Problem ..........................................................................3 1.3 Modeling of the Optimization Problem ............................................5 1.4 Solution with the Graphical Method ................................................ 11 1.5 Convexity ............................................................................................. 13 1.6 Gradient Vector, Directional Derivative, and Hessian Matrix ..... 16 1.7 Linear and Quadratic Approximations ...........................................23 1.8 Organization of the Book ...................................................................25 Chapter Highlights ........................................................................................27 Formulae Chart ..............................................................................................28 Problems ..........................................................................................................29 2. 1-D Optimization Algorithms ...................................................................35 Introduction .........................................................................................35 2.1 2.2 Test Problem ......................................................................................... 37 Solution Techniques ............................................................................38 2.3 2.3.1 Bisection Method ...................................................................38 2.3.2 Newton–Raphson Method ...................................................40 2.3.3 Secant Method ........................................................................42 2.3.4 Cubic Polynomial Fit .............................................................44 2.3.5 Golden Section Method ........................................................46 2.3.6 Other Methods .......................................................................47 2.4 Comparison of Solution Methods .....................................................49 Chapter Highlights ........................................................................................ 51 Formulae Chart .............................................................................................. 52 Problems .......................................................................................................... 52 3. Unconstrained Optimization .....................................................................55 3.1 Introduction .........................................................................................55 3.2 Unidirectional Search ......................................................................... 57 3.3 Test Problem .........................................................................................59 Solution Techniques ............................................................................60 3.4 3.4.1 Steepest Descent Method ...................................................... 62 3.4.2 Newton’s Method ...................................................................63 3.4.3 Modified Newton’s Method .................................................66 3.4.4 Levenberg–Marquardt Method ...........................................66 vii © 2015 by Taylor & Francis Group, LLC

viii Contents 3.4.5 Fletcher–Reeves Conjugate Gradient Method ...................68 3.4.6 DFP Method ............................................................................ 70 3.4.7 BFGS Method ..........................................................................72 3.4.8 Powell Method ....................................................................... 74 3.4.9 Nelder–Mead Algorithm ......................................................75 3.5 Additional Test Functions .................................................................. 78 3.5.1 Rosenbrock Function ............................................................. 78 3.5.2 Quadratic Function ................................................................79 3.5.3 Nonlinear Function ............................................................... 81 3.5.4 Wood’s Function.....................................................................82 3.6 Application to Robotics ......................................................................83 Chapter Highlights ........................................................................................85 Formulae Chart ..............................................................................................86 Problems ..........................................................................................................87 4. Linear Programming ....................................................................................93 Introduction .........................................................................................93 4.1 Solution with the Graphical Method ................................................95 4.2 Standard Form of an LPP ...................................................................98 4.3 4.4 Basic Solution ..................................................................................... 103 Simplex Method ................................................................................ 105 4.5 4.5.1 Multiple Solutions ................................................................ 112 4.5.2 Degeneracy ........................................................................... 114 4.5.3 Two-Phase Method .............................................................. 116 4.5.4 Dual Simplex Method ......................................................... 121 4.6 Interior-Point Method ....................................................................... 125 4.7 Portfolio Optimization ..................................................................... 127 Chapter Highlights ...................................................................................... 131 Formulae Chart ............................................................................................ 133 Problems ........................................................................................................ 133 5. Guided Random Search Methods ........................................................... 139 5.1 Introduction ....................................................................................... 139 5.2 Genetic Algorithms ........................................................................... 140 5.2.1 Initialize Population ............................................................ 142 5.2.2 Fitness Evaluation ................................................................ 143 5.2.3 Reproduction ........................................................................ 143 5.2.4 Crossover and Mutation ..................................................... 147 5.2.5 Multimodal Test Functions ................................................ 148 5.3 Simulated Annealing ........................................................................ 154 5.4 Particle Swarm Optimization .......................................................... 157 5.5 Other Methods .................................................................................. 160 5.5.1 Ant Colony Optimization ................................................... 160 5.5.2 Tabu Search ........................................................................... 163 Chapter Highlights ...................................................................................... 164 © 2015 by Taylor & Francis Group, LLC

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