Optimization Concepts and Applications in Engineering

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1 Optimization Concepts and Applications in Engineering Ashok D. Belegundu, Ph.D. Department of Mechanical Engineering The Pennsylvania State University University Park, Pennsylvania Tirupathi R. Chandrupatia, Ph.D., P.E. Department of Mechanical Engineering Rowan University Glassboro, New Jersey Prentice Hall Upper Saddle River, New Jersey 07458

Contents PREFACE 1 PRELIMINARY CONCEPTS 1.1 Introduction 1 1.2 Historical Sketch 2 1.3 The Nonlinear Programming Problem 3 1.4 Mathematical Fundamentals 6 1.

2 Contents PREFACE 1 PRELIMINARY CONCEPTS 1.1 Introduction Historical Sketch The Nonlinear Programming Problem Mathematical Fundamentals Conclusion 15 Problems 17 Computer Programs 17 2 ONE-DIMENSIONAL UNCONSTRAINED MINIMIZATION 2.1 Introduction Single-Variable Minimization Unimodality and Bracketing the Minimum Fibonacci Method Golden Section Method Polynomial-Based Methods Zero of a Function Conclusion 46 Problems 47 Computer Programs 51

Contents 3 UNCONSTRAINED MINIMIZATION 3.1 Introduction 56 3.2 Necessary and Sufficient Conditions for Optimality 3.3 Convexity 61 3.

3 Contents 3 UNCONSTRAINED MINIMIZATION 3.1 Introduction Necessary and Sufficient Conditions for Optimality 3.3 Convexity Basic Concepts: Starting Design, Direction Vetor, and Step Size The Steepest Descent Method The Conjugate Gradient Method Newton's Method Quasi-Newton Methods Trust Region Methods 81 Problems 83 Computer Programs 86 4 LINEAR PROGRAMMING 4.1 Introduction Linear Programming Problem LP Problems Involving LE (<) Constraints The Simplex Method Treatment of GE and EQ Constraints The Two-Phase Approach, The Big M Method, Revised Simplex Method Duality in Linear Programming The Dual Simplex Method Sensitivity Analysis Interior Approach of Dikin Problem Modeling Quadratic Programming and the Linear Complementary Problem (LCP) Conclusion 128

: Contents vii Problems 128 Computer Programs 128 5 CONSTRAINED MINIMIZATION 141 5.1 Introduction and Problem Formulation 141 5.2 Graphical Solution oftwo-variable Problems 144 5.

4 : Contents vii Problems 128 Computer Programs CONSTRAINED MINIMIZATION Introduction and Problem Formulation Graphical Solution oftwo-variable Problems Necessary Conditions for Optimality Sufficient Conditions for Optimality Convex Problems Sensitivity of Optimum Solution to Problem Parameters Rosen's Gradient Projection Method for Linear Constraints Zoutendijk's Method of Feasible Directions (Nonlinear Constraints) The Generalized Reduced Gradient Method (Nonlinear Constraints) Sequential Quadratic Programming Summary of the Capabilities of Methods for Nonlinear Constrained Problems Solved Examples 189 Problems 195 Computer Programs PENALTY FUNCTION AND DUALITY BASED METHODS Introduction Exterior Penalty Functions Interior Penalty Functions Duality The Augmented Lagrangian Method Duality and Geometrie Programming 242 Problems 248 Computer Programs 251

viii Contents 7 DIRECT SEARCH METHODS FOR NONLINEAR OPTIMIZATION 259 7.1 Introduction 259 7.2 Cyclic Coordinate Search 259 7.3 Hooke and Jeeves Pattern Search Method 262 7.4 Rosenbrock's Method 263 7.

5 viii Contents 7 DIRECT SEARCH METHODS FOR NONLINEAR OPTIMIZATION Introduction Cyclic Coordinate Search Hooke and Jeeves Pattern Search Method Rosenbrock's Method Powell's Method of Conjugate Directions Neider and Mead Simplex Method Simulated Annealing (SA) Genetic Algorithm (GA) Box's Complex Method for Constrained Problems Conclusion 283 Problems 284 Computer Programs INTEGER AND DISCRETE PROGRAMMING Introduction Zero-One Programming Branch and Bound Algorithm for Mixed Integers Gomory Cut Method Farkas' Method for Discrete Nonlinear Monotone Structural Problems Genetic Algorithm for Discrete Programming Conclusion 313 Problems 313 Computer Programs DYNAMIC PROGRAMMING Introduction General Definition of the Dynamic Programming Problem Problem Modeling and Computer Implementation 332

Contents ix 9.4 Discussion and Conclusions 336 Problems 336 Computer Programs 337 10 OPTIMIZATION APPLICATIONS FOR TRANSPORTATION, ASSIGNMENT, AND NETWORK PROBLEMS 340 10.1 Introduction 340 10.

6 Contents ix 9.4 Discussion and Conclusions 336 Problems 336 Computer Programs OPTIMIZATION APPLICATIONS FOR TRANSPORTATION, ASSIGNMENT, AND NETWORK PROBLEMS Introduction Transportation Problem Assignment Problems Network Problems Conclusion 356 Problems 356 Computer Programs PARETO OPTIMALITY 373 il.l Introduction Conceptof Pareto Optimality Generation of the Entire Pareto Curve A Single Best Compromise Pareto Solution 379 Problems FINITE-ELEMENT-BASED OPTIMIZATION Introduction Parameter Optimization Using Gradient Methods-Derivative Calculations Shape Optimization Topology Optimization of Continuum Structures Optimization with Vibration Response Concluding Remarks 412 Problems 415 Computer Programs 419 INDEX 429

Contents. Preface. 1 Introduction Optimization view on mathematical models NLP models, black-box versus explicit expression 3

Contents. Preface. 1 Introduction Optimization view on mathematical models NLP models, black-box versus explicit expression 3 Contents Preface ix 1 Introduction 1 1.1 Optimization view on mathematical models 1 1.2 NLP models, black-box versus explicit expression 3 2 Mathematical modeling, cases 7 2.1 Introduction 7 2.2 Enclosing