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Engineering Optimization
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Contents

Preface xix

Acknowledgments xxvii

Nomenclature xxix

Section 1 Introductory Concepts 1 1 Optimization: Introduction and Concepts 3

1.1 Optimization and Terminology 3

1.2 Optimization Concepts and Definitions 4

1.3 Examples 6

1.4 Terminology Continued 10

1.4.1 Constraint 10

1.4.2 Feasible Solutions 10

1.4.3 Minimize or Maximize 11

1.4.4 Canonical Form of the Optimization Statement 11

1.5 Optimization Procedure 12

1.6 Issues That Shape Optimization Procedures 16

1.7 Opposing Trends 17

1.8 Uncertainty 20

1.9 Over- and Under-specification in Linear Equations 21

1.10 Over- and Under-specification in Optimization 22

1.11 Test Functions 23

1.12 Significant Dates in Optimization 23

1.13 Iterative Procedures 26

1.14 Takeaway 27

1.15 Exercises 27

2 Optimization Application Diversity and Complexity 33

2.1 Optimization 33

2.2 Nonlinearity 33

2.3 Min, Max, Min-Max, Max-Min, ... 34

2.4 Integers and Other Discretization 35

2.5 Conditionals and Discontinuities: Cliffs Ridges/Valleys 36

2.6 Procedures, Not Equations 37

2.7 Static and Dynamic Models 38

2.8 Path Integrals 38

2.9 Economic Optimization and Other Nonadditive Cost Functions 38

2.10 Reliability 39

2.11 Regression 40

2.12 Deterministic and Stochastic 42

2.13 Experimental w.r.t. Modeled OF 43

2.14 Single and Multiple Optima 44

2.16 Inflections 46

2.17 Continuum and Discontinuous DVs 47

2.18 Continuum and Discontinuous Models 47

2.19 Constraints and Penalty Functions 48

2.20 Ranks and Categorization: Discontinuous OFs 50

2.21 Underspecified OFs 51

2.22 Takeaway 51

2.23 Exercises 51

3 Validation: Knowing That the Answer Is Right 53

3.1 Introduction 53

3.2 Validation 53

3.3 Advice on Becoming Proficient 55

3.4 Takeaway 56

3.5 Exercises 57

Section 2 Univariate Search Techniques 59 4 Univariate (Single DV) Search Techniques 61

4.1 Univariate (Single DV) 61

4.2 Analytical Method of Optimization 62

4.2.1 Issues with the Analytical Approach 63

4.3 Numerical Iterative Procedures 64

4.3.1 Newton's Methods 64

4.3.2 Successive Quadratic (A Surrogate Model or Approximating Model Method) 68

4.4 Direct Search Approaches 70

4.4.1 Bisection Method 70

4.4.2 Golden Section Method 72

4.4.3 Perspective at This Point 74

4.4.4 Heuristic Direct Search 74

4.4.5 Leapfrogging 76

4.4.6 LF for Stochastic Functions 79

4.5 Perspectives on Univariate Search Methods 82

4.6 Evaluating Optimizers 85

4.7 Summary of Techniques 85

4.7.1 Analytical Method 86

4.7.2 Newton's (and Variants Like Secant) 86

4.7.4 Golden Section Method 86

4.7.5 Heuristic Direct 87

4.7.6 Leapfrogging 87

4.8 Takeaway 87

4.9 Exercises 88

5 Path Analysis 93

5.1 Introduction 93

5.2 Path Examples 93

5.4 Path Distance Integral 97

5.5 Accumulation along a Path 99

5.6 Slope along a Path 101

5.7 Parametric Path Notation 103

5.8 Takeaway 104

5.9 Exercises 104

6 Stopping and Convergence Criteria: 1-D Applications 107

6.1 Stopping versus Convergence Criteria 107

6.2 Determining Convergence 107

6.2.1 Threshold on the OF 108

6.2.2 Threshold on the Change in the OF 108

6.2.3 Threshold on the Change in the DV 108

6.2.4 Threshold on the Relative Change in the DV 109

6.2.5 Threshold on the Relative Change in the OF 109

6.2.6 Threshold on the Impact of the DV on the OF 109

6.2.7 Convergence Based on Uncertainty Caused by the Givens 109

6.2.8 Multiplayer Range 110

6.3 Combinations of Convergence Criteria 111

6.4 Choosing Convergence Threshold Values 112

6.5 Precision 112

6.6 Other Convergence Criteria 113

6.7 Stopping Criteria to End a Futile Search 113

6.7.1 N Iteration Threshold 114

6.7.2 Execution Error 114

6.7.3 Constraint Violation 114

6.8 Choices! 114

6.9 Takeaway 114

6.10 Exercises 115

Section 3 Multivariate Search Techniques 117 7 Multidimension Application Introduction and the Gradient 119

7.1 Introduction 119

7.2 Illustration of Surface and Terms 122

7.3 Some Surface Analysis 123

7.4 Parametric Notation 128

7.5 Extension to Higher Dimension 130

7.6 Takeaway 131

7.7 Exercises 131

8 Elementary Gradient-Based Optimizers: CSLS and ISD 135

8.1 Introduction 135

8.2 Cauchy's Sequential Line Search 135

8.2.1 CSLS with Successive Quadratic 137

8.2.2 CSLS with Newton/Secant 138

8.2.3 CSLS with Golden Section 138

8.2.4 CSLS with Leapfrogging 138

8.2.5 CSLS with Heuristic Direct Search 139

8.2.6 CSLS Commentary 139

8.2.7 CSLS Pseudocode 140

8.2.8 VBA Code for a 2-DV Application 141

8.3 Incremental Steepest Descent 144

8.3.1 Pseudocode for the ISD Method 144

8.3.2 Enhanced ISD 145

8.3.3 ISD Code 148

8.4 Takeaway 149

8.5 Exercises 149

9 Second-Order Model-Based Optimizers: SQ and NR 155

9.1 Introduction 155

9.2.1 Multivariable SQ 156

9.2.2 SQ Pseudocode 159

9.3 Newton-Raphson 159

9.3.1 NR Pseudocode 162

9.3.2 Attenuate NR 163

9.3.3 Quasi-Newton 166

9.4 Perspective on CSLS, ISD, SQ, and NR 168

9.5 Choosing Step Size for Numerical Estimate of Derivatives 169

9.6 Takeaway 170

9.7 Exercises 170

10 Gradient-Based Optimizer Solutions: LM, RLM, CG, BFGS, RG, and GRG 173

10.1 Introduction 173

10.2 Levenberg-Marquardt (LM) 173

10.2.1 LM VBA Code for a 2-DV Case 175

10.2.2 Modified LM (RLM) 176

10.2.3 RLM Pseudocode 177

10.2.4 RLM VBA Code for a 2-DV Case 178

10.3 Scaled Variables 180

10.5 Broyden-Fletcher-Goldfarb-Shanno (BFGS) 183

10.6 Generalized Reduced Gradient (GRG) 184

10.7 Takeaway 186

10.8 Exercises 186

11 Direct Search Techniques 187

11.1 Introduction 187

11.2 Cyclic Heuristic Direct (CHD) Search 188

11.2.1 CHD Pseudocode 188

11.2.2 CHD VBA Code 189

11.3 Hooke-Jeeves (HJ) 192

11.3.1 HJ Code in VBA 195

11.4 Compare and Contrast CHD and HJ Features: A Summary 197

11.5 Nelder-Mead (NM) Simplex: Spendley, Hext, and Himsworth 199

11.6 Multiplayer Direct Search Algorithms 200

11.7 Leapfrogging 201

11.7.1 Convergence Criteria 208

11.7.2 Stochastic Surfaces 209

11.7.3 Summary 209

11.8 Particle Swarm Optimization 209

11.8.1 Individual Particle Behavior 210

11.8.2 Particle Swarm 213

11.8.3 PSO Equation Analysis 215

11.9 Complex Method (CM) 216

11.10 A Brief Comparison 217

11.11 Takeaway 218

11.12 Exercises 219

12 Linear Programming 223

12.1 Introduction 223

12.2 Visual Representation and Concepts 225

12.3 Basic LP Procedure 228

12.4 Canonical LP Statement 228

12.5 LP Algorithm 229

12.6 Simplex Tableau 230

12.7 Takeaway 231

12.8 Exercises 231

13 Dynamic Programming 233

13.1 Introduction 233

13.2 Conditions 236

13.3 DP Concept 237

13.4 Some Calculation Tips 240

13.5 Takeaway 241

13.6 Exercises 241

14 Genetic Algorithms and Evolutionary Computation 243

14.1 Introduction 243

14.2 GA Procedures 243

14.3 Fitness of Selection 245

14.4 Takeaway 250

14.5 Exercises 250

15 Intuitive Optimization 253

15.1 Introduction 253

15.2 Levels 254

15.3 Takeaway 254

15.4 Exercises 254

16 Surface Analysis II 257

16.1 Introduction 257

16.2 Maximize Is Equivalent to Minimize the Negative 257

16.3 Scaling by a Positive Number Does Not Change DV 258

16.4 Scaled and Translated OFs Do Not Change DV 258

16.5 Monotonic Function Transformation Does Not Change DV 258

16.6 Impact on Search Path or NOFE 261

16.7 Inequality Constraints 263

16.8 Transforming DVs 263

16.9 Takeaway 263

16.10 Exercises 263

17 Convergence Criteria 2: N-D Applications 265

17.1 Introduction 265

17.2 Defining an Iteration 265

17.3 Criteria for Single TS Deterministic Procedures 266

17.4 Criteria for Multiplayer Deterministic Procedures 267

17.5 Stochastic Applications 268

17.7 Takeaway 269

17.8 Exercises 269

18 Enhancements to Optimizers 271

18.1 Introduction 271

18.2 Criteria for Replicate Trials 271

18.3 Quasi-Newton 274

18.4 Coarse-Fine Sequence 275

18.5 Number of Players 275

18.7 Adjustment of Optimizer Coefficient Values or Options in Process 276

18.8 Initialization Range 277

18.9 OF and DV Transformations 277

18.10 Takeaway 278

18.11 Exercises 278

Section 4 Developing Your Application Statements 279 19 Scaled Variables and Dimensional Consistency 281

19.1 Introduction 281

19.2 A Scaled Variable Approach 283

19.3 Sampling of Issues with Primitive Variables 283

19.4 Linear Scaling Options 285

19.5 Nonlinear Scaling 286

19.6 Takeaway 287

19.7 Exercises 287

20 Economic Optimization 289

20.1 Introduction 289

20.2 Annual Cash Flow 290

20.3 Including Risk as an Annual Expense 291

20.4 Capital 293

20.5 Combining Capital and Nominal Annual Cash Flow 293

20.6 Combining Time Value and Schedule of Capital and Annual Cash Flow 296

20.7 Present Value 297

20.8 Including Uncertainty 298

20.8.1 Uncertainty Models 301

20.8.2 Methods to Include Uncertainty in an Optimization 303

20.9 Takeaway 304

20.10 Exercises 304

21 Multiple OF and Constraint Applications 305

21.1 Introduction 305

21.2 Solution 1: Additive Combinations of the Functions 306

21.2.1 Solution 1a: Classic Weighting Factors 307

21.2.2 Solution 1b: Equal Concern Weighting 307

21.2.3 Solution 1c: Nonlinear Weighting 309

21.3 Solution 2: Nonadditive OF Combinations 311

21.4 Solution 3: Pareto Optimal 311

21.5 Takeaway 316

21.6 Exercises 316

22 Constraints 319

22.1 Introduction 319

22.2 Equality Constraints 320

22.2.1 Explicit Equality Constraints 320

22.2.2 Implicit Equality Constraints 321

22.3 Inequality Constraints 321

22.3.1 Penalty Function: Discontinuous 323

22.3.2 Penalty Function: Soft Constraint 323

22.3.3 Inequality Constraints: Slack and Surplus Variables 325

22.4 Constraints: Pass/Fail Categories 329

22.5 Hard Constraints Can Block Progress 330

22.7 Constraint-Equivalent Features 332

22.8 Takeaway 332

22.9 Exercises 332

23 Multiple Optima 335

23.1 Introduction 335

23.2 Solution: Multiple Starts 337

23.2.1 A Priori Method 340

23.2.2 A Posteriori Method 342

23.2.3 Snyman and Fatti Criterion A Posteriori Method 345

23.3 Other Options 348

23.4 Takeaway 349

23.5 Exercises 350

24 Stochastic Objective Functions 353

24.1 Introduction 353

24.2 Method Summary for Optimizing Stochastic Functions 356

24.2.1 Step 1: Replicate the Apparent Best Player 356

24.2.2 Step 2: Steady-State Detection 357

24.3 What Value to Report? 358

24.4 Application Examples 359

24.4.1 GMC Control of Hot and Cold Mixing 359

24.4.2 MBC of Hot and Cold Mixing 359

24.4.3 Batch Reaction Management 359

24.4.4 Reservoir and Stochastic Boot Print 361

24.4.5 Optimization Results 362

24.5 Takeaway 365

24.6 Exercises 365

25 Effects of Uncertainty 367

25.1 Introduction 367

25.2 Sources of Error and Uncertainty 368

25.3 Significant Digits 370

25.4 Estimating Uncertainty on Values 371

25.5 Propagating Uncertainty on DV Values 372

25.5.1 Analytical Method 373

25.5.2 Numerical Method 375

25.6 Implicit Relations 378

25.7 Estimating Uncertainty in DV and OF 378

25.8 Takeaway 379

25.9 Exercises 379

26 Optimization of Probable Outcomes and Distribution Characteristics 381

26.1 Introduction 381

26.2 The Concept of Modeling Uncertainty 385

26.3 Stochastic Approach 387

26.4 Takeaway 389

26.5 Exercises 389

27 Discrete and Integer Variables 391

27.1 Introduction 391

27.2 Optimization Solutions 394

27.2.1 Exhaustive Search 394

27.2.2 Branch and Bound 394

27.2.3 Cyclic Heuristic 394

27.2.4 Leapfrogging or Other Multiplayer Search 395

27.3 Convergence 395

27.4 Takeaway 395

27.5 Exercises 395

28 Class Variables 397

28.1 Introduction 397

28.2 The Random Keys Method: Sequence 398

28.3 The Random Keys Method: Dichotomous Variables 400

28.5 Takeaway 401

28.6 Exercises 401

29 Regression 403

29.1 Introduction 403

29.2 Perspective 404

29.3 Least Squares Regression: Traditional View on Linear Model Parameters 404

29.4 Models Nonlinear in DV 405

29.4.1 Models with a Delay 407

29.5 Maximum Likelihood 408

29.5.1 Akaho's Method 411

29.6 Convergence Criterion 416

29.7 Model Order or Complexity 421

29.8 Bootstrapping to Reveal Model Uncertainty 425

29.8.1 Interpretation of Bootstrapping Analysis 428

29.8.2 Appropriating Bootstrapping 430

29.9 Perspective 431

29.10 Takeaway 431

29.11 Exercises 432

Section 5 Perspective on Many Topics 441 30 Perspective 443

30.1 Introduction 443

30.2 Classifications 443

30.3 Elements Associated with Optimization 445

30.4 Root Finding Is Not Optimization 446

30.5 Desired Engineering Attributes 446

30.6 Overview of Optimizers and Attributes 447

30.6.1 Gradient Based: Cauchy Sequential Line Search, Incremental Steepest Descent, GRG, Etc. 447

30.6.2 Local Surface Characterization Based: Newton-Raphson, Levenberg-Marquardt, Successive Quadratic, RLM, Quasi-Newton, Etc. 448

30.6.3 Direct Search with Single Trial Solution: Cyclic Heuristic, Hooke-Jeeves, and Nelder-Mead 448

30.6.4 Multiplayer Direct Search Optimizers: Leapfrogging, Particle Swarm, and Genetic Algorithms 448

30.7 Choices 448

30.8 Variable Classifications 449

30.8.1 Nominal 449

30.8.2 Ordinal 450

30.8.3 Cardinal 450

30.9 Constraints 451

30.10 Takeaway 453

30.11 Exercises 453

31 Response Surface Aberrations 459

31.1 Introduction 459

31.2 Cliffs (Vertical Walls) 459

31.3 Sharp Valleys (or Ridges) 459

31.4 Striations 463

31.5 Level Spots (Functions 1, 27, 73, 84) 463

31.6 Hard-to-Find Optimum 466

31.7 Infeasible Calculations 468

31.8 Uniform Minimum 468

31.9 Noise: Stochastic Response 469

31.10 Multiple Optima 471

31.11 Takeaway 473

31.12 Exercises 473

32 Identifying the Models, OF, DV, Convergence Criteria, and Constraints 475

32.1 Introduction 475

32.2 Evaluate the Results 476

32.3 Takeaway 482

32.4 Exercises 482

33 Evaluating Optimizers 489

33.1 Introduction 489

33.2 Challenges to Optimizers 490

33.3 Stakeholders 490

33.4 Metrics of Optimizer Performance 490

33.5 Designing an Experimental Test 492

33.6 Takeaway 495

33.7 Exercises 496

34 Troubleshooting Optimizers 499

34.1 Introduction 499

34.2 DV Values Do Not Change 499

34.3 Multiple DV Values for the Same OF Value 499

34.4 EXE Error 500

34.5 Extreme Values 500

34.6 DV Is Dependent on Convergence Threshold 500

34.7 OF Is Irreproducible 501

34.8 Concern over Results 501

34.9 CDF Features 501

34.10 Parameter Correlation 502

34.11 Multiple Equivalent Solutions 504

34.12 Takeaway 504

34.13 Exercises 504

Section 6 Analysis of Leapfrogging Optimization 505 35 Analysis of Leapfrogging 507

35.1 Introduction 507

35.2 Balance in an Optimizer 508

35.3 Number of Initializations to be Confident That the Best Will Draw All Others to the Global Optimum 510

35.3.1 Methodology 511

35.3.2 Experimental 512

35.3.3 Results 513

35.4 Leap-To Window Amplification Analysis 515

35.5 Analysis of and M to Prevent Convergence on the Side of a Hill 519

35.6 Analysis of and M to Minimize NOFE 521

35.7 Probability Distribution of Leap-Overs 522

35.7.1 Data 526

35.8 Takeaway 527

35.9 Exercises 528

Section 7 Case Studies 529 36 Case Study 1: Economic Optimization of a Pipe System 531

36.1 Process and Analysis 531

36.1.1 Deterministic Continuum Model 531

36.1.2 Deterministic Discontinuous Model 534

36.1.3 Stochastic Discontinuous Model 535

36.2 Exercises 536

37 Case Study 2: Queuing Study 539

37.1 The Process and Analysis 539

37.2 Exercises 541

38 Case Study 3: Retirement Study 543

38.1 The Process and Analysis 543

38.2 Exercises 550

39 Case Study 4: A Goddard Rocket Study 551

39.1 The Process and Analysis 551

39.2 Pre-Assignment Note 554

39.3 Exercises 555

40 Case Study 5: Reservoir 557

40.1 The Process and Analysis 557

40.2 Exercises 559

41 Case Study 6: Area Coverage 561

41.1 Description and Analysis 561

41.2 Exercises 562

42 Case Study 7: Approximating Series Solution to an ODE 565

42.1 Concepts and Analysis 565

42.2 Exercises 568

43 Case Study 8: Horizontal Tank Vapor-Liquid Separator 571

43.1 Description and Analysis 571

43.2 Exercises 576

44 Case Study 9: In Vitro Fertilization 579

44.1 Description and Analysis 579

44.2 Exercises 583

45 Case Study 10: Data Reconciliation 585

45.1 Description and Analysis 585

45.2 Exercises 588

Section 8 Appendices 591

Appendix A Mathematical Concepts and Procedures 593

Appendix B Root Finding 605

Appendix C Gaussian Elimination 611

Appendix D Steady-State Identification in Noisy Signals 621

Appendix E Optimization Challenge Problems (2-D and Single OF) 635

Appendix F Brief on VBA Programming: Excel in Office 2013 709

Section 9 References and Index 717