In this revised and enhanced second edition of Optimization Concepts and.
Applications in Engineering, the already robust pedagogy has been enhanced
with ...
Cambridge University Press 978-0-521-87846-3 - Optimization Concepts and Applications in Engineering: Second Edition Ashok D. Belegundu and Tirupathi R. Chandrupatla Frontmatter More information
OPTIM IZATION CONCEPTS AND APPLICATIONS IN ENGINEERING Second Edition
It is vitally important to meet or exceed previous quality and reliability standards while at the same time reducing resource consumption. This textbook addresses this critical imperative integrating theory, modeling, the development of numerical methods, and problem solving, thus preparing the student to apply optimization to real-world problems. This text covers a broad variety of optimization problems using the following: unconstrained, constrained, gradient, and nongradient techniques; duality concepts; multiobjective optimization; linear, integer, geometric, and dynamic programming with applications; and finite element-based optimization. In this revised and enhanced second edition of Optimization Concepts and Applications in Engineering, the already robust pedagogy has been enhanced with more detailed explanations and an increased number of solved examples and end-of-chapter problems. The source codes are now available free on multiple platforms. It is ideal for advanced undergraduate or graduate courses and for practicing engineers in all engineering disciplines, as well as in applied mathematics. Ashok D. Belegundu has been a Professor of Mechanical Engineering at The Pennsylvania State University, University Park, since 1986. Prior to this, he taught at GMI, now Kettering University, in Michigan. He received his B. Tech. degree from IIT Madras and his Ph.D. from the University of Iowa. He has been a principal investigator on research projects involving optimization for several agencies including the National Science Foundation, Army Research Office, NASA, SERC (UK), MacNeal-Schwendler Corporation, Gentex Corporation, and Ingersoll-Rand. He has organized two international conferences on optimization in industry and has authored or edited four books and written a chapter in a book. A detailed list of his publications and projects can be found at http://www.mne.psu.edu/Directories/Faculty/Belegundu-A.html. He has advised more than 50 graduate students. He has given short courses on finite elements and optimization to the Forging Industry Association, Hazleton Pumps, and Infosys (India). He has served as an associate editor for AIAA Journal and for Mechanics Based Design of Structures and Machines. He teaches a distance education course on optimal design through Penn State. Tirupathi R. Chandrupatla has been a Professor and Founding Chair of Mechanical Engineering at Rowan University since 1995. He started his career as a design engineer with Hindustan Machine Tools (HMT), Bangalore. He then taught at IIT Bombay. Professor Chandrupatla also taught at the University of Kentucky, Lexington, and GMI Engineering and Management Institute (now Kettering University), before joining Rowan. He received the Lindback Distinguished Teaching Award at Rowan University in 2005. He is also the author of Quality and Reliability in Engineering and two books on finite element analysis. Professor Chandrupatla has broad research interests, which include design, optimization, manufacturing engineering, finite element analysis, and quality and reliability. He has published widely in these areas and serves as an industry consultant. Professor Chandrupatla is a registered Professional Engineer and also a Certified Manufacturing Engineer. He is a member of ASEE, ASME, SAE, and SME. For further information, visit http://users.rowan.edu/∼chandrupatla.
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Cambridge University Press 978-0-521-87846-3 - Optimization Concepts and Applications in Engineering: Second Edition Ashok D. Belegundu and Tirupathi R. Chandrupatla Frontmatter More information
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Cambridge University Press 978-0-521-87846-3 - Optimization Concepts and Applications in Engineering: Second Edition Ashok D. Belegundu and Tirupathi R. Chandrupatla Frontmatter More information
Optimization Concepts and Applications in Engineering Second Edition Ashok D. Belegundu The Pennsylvania State University
Tirupathi R. Chandrupatla Rowan University
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Cambridge University Press 978-0-521-87846-3 - Optimization Concepts and Applications in Engineering: Second Edition Ashok D. Belegundu and Tirupathi R. Chandrupatla Frontmatter More information
cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, ˜ Paulo, Delhi, Mexico City Singapore, Sao Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9780521878463 C Ashok D. Belegundu and Tirupathi R. Chandrupatla 1999, 2011
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 by Prentice Hall 1999 Second edition published 2011 Reprinted 2012 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication data Belegundu, Ashok D., 1956– Optimization concepts and applications in engineering / Ashok D. Belegundu, Tirupathi R. Chandrupatla. – 2nd ed. p. cm. Includes index. ISBN 978-0-521-87846-3 (hardback) 1. Engineering – Mathematical models. 2. Engineering design – Mathematics. 3. Mathematical optimization. 4. Engineering models. I. Chandrupatla, Tirupathi R., 1944– II. Title. TA342.B45 2011 519.602 462–dc22 2010049376 ISBN 978-0-521-87846-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.
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Cambridge University Press 978-0-521-87846-3 - Optimization Concepts and Applications in Engineering: Second Edition Ashok D. Belegundu and Tirupathi R. Chandrupatla Frontmatter More information
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Cambridge University Press 978-0-521-87846-3 - Optimization Concepts and Applications in Engineering: Second Edition Ashok D. Belegundu and Tirupathi R. Chandrupatla Frontmatter More information
Contents
Preface
page xi
1 Preliminary Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Introduction Historical Sketch The Nonlinear Programming Problem Optimization Problem Modeling Graphical Solution of One- and Two-Variable Problems Existence of a Minimum and a Maximum: Weierstrass Theorem Quadratic Forms and Positive Definite Matrices C n Continuity of a Function Gradient Vector, Hessian Matrix, and Their Numerical Evaluation Using Divided Differences 1.10 Taylor’s Theorem, Linear, and Quadratic Approximations 1.11 Miscellaneous Topics
1 2 4 7 19 22 25 26 28 33 36
2 One-Dimensional Unconstrained Minimization . . . . . . . . . . . . . . . . . . 46 2.1 2.2 2.3 2.4 2.5 2.6 2.7
Introduction Theory Related to Single Variable (Univariate) Minimization Unimodality and Bracketing the Minimum Fibonacci Method Golden Section Method Polynomial-Based Methods Shubert–Piyavskii Method for Optimization of Non-unimodal Functions 2.8 Using MATLAB 2.9 Zero of a Function
46 46 54 55 63 67 75 77 78 vii
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3 Unconstrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.1 3.2 3.3 3.4
Introduction Necessary and Sufficient Conditions for Optimality Convexity Basic Concepts: Starting Design, Direction Vector, and Step Size 3.5 The Steepest Descent Method 3.6 The Conjugate Gradient Method 3.7 Newton’s Method 3.8 Quasi-Newton Methods 3.9 Approximate Line Search 3.10 Using MATLAB
89 90 94 96 99 106 112 116 121 123
4 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.1 Introduction 4.2 Linear Programming Problem 4.3 Problem Illustrating Modeling, Solution, Solution Interpretation, and Lagrange Multipliers 4.4 Problem Modeling 4.5 Geometric Concepts: Hyperplanes, Halfspaces, Polytopes, Extreme Points 4.6 Standard form of an LP 4.7 The Simplex Method – Starting with LE (≤) Constraints 4.8 Treatment of GE and EQ Constraints 4.9 Revised Simplex Method 4.10 Duality in Linear Programming 4.11 The Dual Simplex Method 4.12 Sensitivity Analysis 4.13 Interior Approach 4.14 Quadratic Programming (QP) and the Linear Complementary Problem (LCP)
131 131 132 137 142 144 146 152 157 161 163 166 172 176
5 Constrained Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 5.1 5.2 5.3 5.4 5.5 5.6
Introduction Graphical Solution of Two-Variable Problems Use of EXCEL SOLVER and MATLAB Formulation of Problems in Standard NLP Form Necessary Conditions for Optimality Sufficient Conditions for Optimality
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Cambridge University Press 978-0-521-87846-3 - Optimization Concepts and Applications in Engineering: Second Edition Ashok D. Belegundu and Tirupathi R. Chandrupatla Frontmatter More information
Contents
5.7 Convexity 5.8 Sensitivity of Optimum Solution to Problem Parameters 5.9 Rosen’s Gradient Projection Method for Linear Constraints 5.10 Zoutendijk’s Method of Feasible Directions (Nonlinear Constraints) 5.11 The Generalized Reduced Gradient Method (Nonlinear Constraints) 5.12 Sequential Quadratic Programming (SQP) 5.13 Features and Capabilities of Methods Presented in this Chapter
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212 214 216 222 232 241 247
6 Penalty Functions, Duality, and Geometric Programming . . . . . . . . . 261 6.1 6.2 6.3 6.4 6.5 6.6
Introduction Exterior Penalty Functions Interior Penalty Functions Duality The Augmented Lagrangian Method Geometric Programming
261 261 267 269 276 281
7 Direct Search Methods for Nonlinear Optimization . . . . . . . . . . . . . . 294 7.1 Introduction 7.2 Cyclic Coordinate Search 7.3 Hooke and Jeeves Pattern Search Method 7.4 Rosenbrock’s Method 7.5 Powell’s Method of Conjugate Directions 7.6 Nelder and Mead Simplex Method 7.7 Simulated Annealing (SA) 7.8 Genetic Algorithm (GA) 7.9 Differential Evolution (DE) 7.10 Box’s Complex Method for Constrained Problems
294 294 298 301 304 307 314 318 324 325
8 Multiobjective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 8.1 8.2 8.3 8.4
Introduction Concept of Pareto Optimality Generation of the Entire Pareto Curve Methods to Identify a Single Best Compromise Solution
338 339 343 345
9 Integer and Discrete Programming . . . . . . . . . . . . . . . . . . . . . . . . . . 359 9.1 Introduction 9.2 Zero–One Programming 9.3 Branch and Bound Algorithm for Mixed Integers (LP-Based)
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Cambridge University Press 978-0-521-87846-3 - Optimization Concepts and Applications in Engineering: Second Edition Ashok D. Belegundu and Tirupathi R. Chandrupatla Frontmatter More information
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9.4 Gomory Cut Method 9.5 Farkas’ Method for Discrete Nonlinear Monotone Structural Problems 9.6 Genetic Algorithm for Discrete Programming
372 377 380
10 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 10.1 Introduction 10.2 The Dynamic Programming Problem and Approach 10.3 Problem Modeling and Computer Implementation
385 387 392
11 Optimization Applications for Transportation, Assignment, and Network Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 11.1 Introduction 11.2 Transportation Problem 11.3 Assignment Problems 11.4 Network Problems
400 400 408 413
12 Finite Element-Based Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 424 12.1 Introduction 12.2 Derivative Calculations 12.3 Sizing (i.e., Parameter) Optimization via Optimality Criteria and Nonlinear Programming Methods 12.4 Topology Optimization of Continuum Structures 12.5 Shape Optimization 12.6 Optimization with Dynamic Response Index
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Cambridge University Press 978-0-521-87846-3 - Optimization Concepts and Applications in Engineering: Second Edition Ashok D. Belegundu and Tirupathi R. Chandrupatla Frontmatter More information
Preface
This book is a revised and enhanced edition of the first edition. The authors have identified a clear need for teaching engineering optimization in a manner that integrates theory, algorithms, modeling, and hands-on experience based on their extensive experience in teaching, research, and interactions with students. They have strived to adhere to this pedagogy and reinforced it further in the second edition, with more detailed explanations, an increased number of solved examples and endof-chapter problems, and source codes on multiple platforms. The development of the software, which parallels the theory, has helped to explain the implementation aspects in the text with greater insight and accuracy. Students have integrated the optimization programs with simulation codes in their theses. The programs can be tried out by researchers and practicing engineers as well. Programs on the CD-ROM have been developed in Matlab, Excel VBA, VBScript, and Fortran. A battery of methods is available for the user. This leads to effective solution of problems since no single method can be successful on all problems. The book deals with a variety of optimization problems: unconstrained, constrained, gradient, and nongradient techniques; duality concepts; multiobjective optimization; linear, integer, geometric, and dynamic programming with applications; and finite element–based optimization. Matlab graphics and optimization toolbox routines and the Excel Solver optimizer are presented in detail. Through solved examples, problem-solving strategies are presented for handling problems where the number of variables depends on the number of discretization points in a mesh and for handling time-dependent constraints. Chapter 8 deals exclusively with treatment of the objective function itself as opposed to methods for minimizing it. This book can be used in courses at the graduate or senior-undergraduate level and as a learning resource for practicing engineers. Specifically, the text can be used xi
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Cambridge University Press 978-0-521-87846-3 - Optimization Concepts and Applications in Engineering: Second Edition Ashok D. Belegundu and Tirupathi R. Chandrupatla Frontmatter More information
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Preface
in courses on engineering optimization, design optimization, structural optimization, and nonlinear programming. The book may be used in mechanical, aerospace, civil, industrial, architectural, chemical, and electrical engineering, as well as in applied mathematics. In deciding which chapters are to be covered in a course, the instructor may note the following. Chapters 1, 2, 3.1–3.5, and 8 are fundamental. Chapters 4, 9, and 11 focus on linear problems, whereas Chapters 5–7 focus on nonlinear problems. Even if the focus is on nonlinear problems, Sections 4.1–4.6 present important concepts related to constraints. Chapters 10 and 12 are specialized topics. Thus, for instance, a course on structural optimization (i.e., finite element-based optimization) may cover Chapters 1–2, 3.1–3.5, 4.1–4.6, 5, 6, 7.7–7.10, 8, and 12. We are grateful to the students at our respective institutions for motivating us to develop this book. It has been a pleasure working with our editor, Peter Gordon.
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Cambridge University Press 978-0-521-87846-3 - Optimization Concepts and Applications in Engineering: Second Edition Ashok D. Belegundu and Tirupathi R. Chandrupatla Frontmatter More information
OPTIM IZATION CONCEPTS AND APPLICATIONS IN ENGINEERING Second Edition
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