optimization concepts and applications in …...optimization concepts and applications in...
TRANSCRIPT
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
1.5 Conclusion 15
Problems 17
Computer Programs 17
2 ONE-DIMENSIONAL UNCONSTRAINED MINIMIZATION
2.1 Introduction 21
2.2 Single-Variable Minimization 21
2.3 Unimodality and Bracketing the Minimum 28
2.4 Fibonacci Method 30
2.5 Golden Section Method 36
2.6 Polynomial-Based Methods 39
2.7 Zero of a Function 43
2.8 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
3.4 Basic Concepts: Starting Design, Direction Vetor, and Step Size 63
3.5 The Steepest Descent Method 64
3.6 The Conjugate Gradient Method 70
3.7 Newton's Method 74
3.8 Quasi-Newton Methods 78
3.9 Trust Region Methods 81
Problems 83
Computer Programs 86
4 LINEAR PROGRAMMING
4.1 Introduction 92
4.2 Linear Programming Problem 92
4.3 LP Problems Involving LE (<) Constraints 94
4.4 The Simplex Method 96
4.5 Treatment of GE and EQ Constraints 100 4.5.1 The Two-Phase Approach, 101
4.5.2 The Big M Method, 104
4.6 Revised Simplex Method 105
4.7 Duality in Linear Programming 108
4.8 The Dual Simplex Method 110
4.9 Sensitivity Analysis 113
4.10 Interior Approach of Dikin 116
4.11 Problem Modeling 119 4.12 Quadratic Programming and the Linear
Complementary Problem (LCP) 125
4.13 Conclusion 128
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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
5.3 Necessary Conditions for Optimality 146
5.4 Sufficient Conditions for Optimality 157
5.5 Convex Problems 159
5.6 Sensitivity of Optimum Solution to Problem Parameters 161
5.7 Rosen's Gradient Projection Method for Linear Constraints 163
5.8 Zoutendijk's Method of Feasible Directions (Nonlinear Constraints) 168
5.9 The Generalized Reduced Gradient Method (Nonlinear Constraints) 176
5.10 Sequential Quadratic Programming 183
5.11 Summary of the Capabilities of Methods for Nonlinear Constrained Problems 189
5.12 Solved Examples 189
Problems 195
Computer Programs 203
6 PENALTY FUNCTION AND DUALITY BASED METHODS 222
6.1 Introduction 222
6.2 Exterior Penalty Functions 222
6.3 Interior Penalty Functions 227
6.4 Duality 229
6.5 The Augmented Lagrangian Method 235
6.6 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 Powell's Method of Conjugate Directions 265
7.6 Neider and Mead Simplex Method 266
7.7 Simulated Annealing (SA) 272
7.8 Genetic Algorithm (GA) 276
7.9 Box's Complex Method for Constrained Problems 279
7.10 Conclusion 283
Problems 284
Computer Programs 286
8 INTEGER AND DISCRETE PROGRAMMING 296
8.1 Introduction 296
8.2 Zero-One Programming 298
8.3 Branch and Bound Algorithm for Mixed Integers 303
8.4 Gomory Cut Method 306
8.5 Farkas' Method for Discrete Nonlinear Monotone Structural Problems 310
8.6 Genetic Algorithm for Discrete Programming 313
8.7 Conclusion 313
Problems 313
Computer Programs 316
9 DYNAMIC PROGRAMMING 327
9.1 Introduction 327
9.2 General Definition of the Dynamic Programming Problem 329
9.3 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
10.2 Transportation Problem 340
10.3 Assignment Problems 347
10.4 Network Problems 351
10.5 Conclusion 356
Problems 356
Computer Programs 359
11 PARETO OPTIMALITY 373
i l . l Introduction 373
11.2 Conceptof Pareto Optimality 374
11.3 Generation of the Entire Pareto Curve 377
11.4 A Single Best Compromise Pareto Solution 379
Problems 382
12 FINITE-ELEMENT-BASED OPTIMIZATION 385
12.1 Introduction 385
12.2 Parameter Optimization Using Gradient Methods-Derivative Calculations 386
12.3 Shape Optimization 395
12.4 Topology Optimization of Continuum Structures 402
12.5 Optimization with Vibration Response 408
12.6 Concluding Remarks 412
Problems 415
Computer Programs 419
INDEX 429