Optimization Concepts and Applications in Engineering, Second Edition

CONTENTS......Page 9
Preface......Page 13
1.1 Introduction......Page 17
1.2 Historical Sketch......Page 18
1.3 The Nonlinear Programming Problem......Page 20
1.4 Optimization Problem Modeling......Page 23
1.5 Graphical Solution of One- and Two-Variable Problems......Page 35
1.6 Existence of a Minimum and a Maximum: Weierstrass Theorem......Page 38
1.7 Quadratic Forms and Positive Definite Matrices......Page 41
1.8 C n Continuity of a Function......Page 42
1.9 Gradient Vector, Hessian Matrix, and Their Numerical
Evaluation Using Divided Differences......Page 44
1.10 Taylor’s Theorem, Linear, and Quadratic Approximations......Page 49
1.11 Miscellaneous Topics......Page 52
2.2 Theory Related to Single Variable (Univariate) Minimization......Page 62
2.3 Unimodality and Bracketing the Minimum......Page 70
2.4 Fibonacci Method......Page 71
2.5 Golden Section Method......Page 79
2.6 Polynomial-Based Methods......Page 83
2.7 Shubert–Piyavskii Method for Optimization of Non-unimodal
Functions......Page 91
2.8 Using MATLAB......Page 93
2.9 Zero of a Function......Page 94
3.1 Introduction......Page 105
3.2 Necessary and Sufficient Conditions for Optimality......Page 106
3.3 Convexity......Page 110
3.4 Basic Concepts: Starting Design, Direction Vector, and
Step Size......Page 112
3.5 The Steepest Descent Method......Page 115
3.6 The Conjugate Gradient Method......Page 122
3.7 Newton’s Method......Page 128
3.8 Quasi-Newton Methods......Page 132
3.9 Approximate Line Search......Page 137
3.10 Using MATLAB......Page 139
4.2 Linear Programming Problem......Page 147
4.3 Problem Illustrating Modeling, Solution, Solution
Interpretation, and Lagrange Multipliers......Page 148
4.4 Problem Modeling......Page 153
4.5 Geometric Concepts: Hyperplanes, Halfspaces, Polytopes,
Extreme Points......Page 158
4.6 Standard form of an LP......Page 160
4.7 The Simplex Method – Starting with LE (≤) Constraints......Page 162
4.8 Treatment of GE and EQ Constraints......Page 168
4.9 Revised Simplex Method......Page 173
4.10 Duality in Linear Programming......Page 177
4.11 The Dual Simplex Method......Page 179
4.12 Sensitivity Analysis......Page 182
4.13 Interior Approach......Page 188
4.14 Quadratic Programming (QP) and the Linear Complementary Problem (LCP)......Page 192
5.1 Introduction......Page 205
5.2 Graphical Solution of Two-Variable Problems......Page 208
5.3 Use of EXCEL SOLVER and MATLAB......Page 209
5.4 Formulation of Problems in Standard NLP Form......Page 211
5.5 Necessary Conditions for Optimality......Page 213
5.6 Sufficient Conditions for Optimality......Page 225
5.7 Convexity......Page 228
5.8 Sensitivity of Optimum Solution to Problem Parameters......Page 230
5.9 Rosen’s Gradient Projection Method for Linear Constraints......Page 232
5.10 Zoutendijk's Method of Feasible Directions (Nonlinear Constraints)......Page 238
5.11 The Generalized Reduced Gradient Method (Nonlinear Constraints)......Page 248
5.12 Sequential Quadratic Programming (SQP)......Page 257
5.13 Features and Capabilities of Methods Presented in this Chapter......Page 263
6.2 Exterior Penalty Functions......Page 277
6.3 Interior Penalty Functions......Page 283
6.4 Duality......Page 285
6.5 The Augmented Lagrangian Method......Page 292
6.6 Geometric Programming......Page 297
7.2 Cyclic Coordinate Search......Page 310
7.3 Hooke and Jeeves Pattern Search Method......Page 314
7.4 Rosenbrock’s Method......Page 317
7.5 Powell’s Method of Conjugate Directions......Page 320
7.6 Nelder and Mead Simplex Method......Page 323
7.7 Simulated Annealing (SA)......Page 330
7.8 Genetic Algorithm (GA)......Page 334
7.9 Differential Evolution (DE)......Page 340
7.10 Box’s Complex Method for Constrained Problems......Page 341
8.1 Introduction......Page 354
8.2 Concept of Pareto Optimality......Page 355
8.3 Generation of the Entire Pareto Curve......Page 359
8.4 Methods to Identify a Single Best Compromise Solution......Page 361
9.1 Introduction......Page 375
9.2 Zero–One Programming......Page 377
9.3 Branch and Bound Algorithm for Mixed Integers (LP-Based)......Page 384
9.4 Gomory Cut Method......Page 388
9.5 Farkas’ Method for Discrete Nonlinear Monotone
Structural Problems......Page 393
9.6 Genetic Algorithm for Discrete Programming......Page 396
10.1 Introduction......Page 401
10.2 The Dynamic Programming Problem and Approach......Page 403
10.3 Problem Modeling and Computer Implementation......Page 408
11.2 Transportation Problem......Page 416
11.3 Assignment Problems......Page 424
11.4 Network Problems......Page 429
12.1 Introduction......Page 440
12.2 Derivative Calculations......Page 443
12.3 Sizing (i.e.,Parameter) Optimization via Optimality Criteria and Nonlinear Programming Methods......Page 448
12.4 Topology Optimization of Continuum Structures......Page 453
12.5 Shape Optimization......Page 457
12.6 Optimization with Dynamic Response......Page 465
Index......Page 477