Nonlinear programming

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 14
1.1 Scope of Optimization Problems......Page 18
1.2 Classification of Optimization Problems......Page 20
1.3 Optimization Applications in Chemical Engineering......Page 22
1.4 Nonlinear Programming Examples in Chemical Engineering......Page 23
1.4.1 Design of a Small Heat Exchanger Network......Page 24
1.4.2 Real-Time Optimization of a Distillation Column......Page 26
1.4.3 Model Predictive Control......Page 28
1.5 A Motivating Application......Page 30
1.7 Exercises......Page 32
2.1 Introduction......Page 34
2.2.1 Vectors and Matrices......Page 36
2.2.2 Quadratic Forms......Page 39
2.2.3 Classification of Functions......Page 42
2.3 Optimality Conditions......Page 44
2.4.1 Direct Search Methods......Page 47
2.4.2 Methods That Require Derivatives......Page 50
2.6 Exercises......Page 54
3.1 Introduction......Page 56
3.2 Modification of the Hessian Matrix......Page 57
3.3 Quasi-Newton Methods......Page 59
3.4 Line Search Methods......Page 63
3.5 Trust Region Methods......Page 69
3.5.1 Convex Model Problems......Page 70
3.5.2 Nonconvex Model Problems......Page 73
3.7 Exercises......Page 77
4.1 Introduction......Page 80
4.1.1 Constrained Convex Problems......Page 81
4.2 Local Optimality Conditions-A Kinematic Interpretation......Page 85
4.3 Analysis of KKT Conditions......Page 89
4.3.1 Linearly Constrained Problems......Page 92
4.3.2 Nonlinearly Constrained Problems......Page 93
4.3.3 Second Order Conditions......Page 96
4.4.1 Description of Linear Programming......Page 101
4.4.2 Description of Quadratic Programming......Page 102
4.4.3 Portfolio Planning Case Study......Page 103
4.5 Summary and Notes for Further Reading......Page 106
4.6 Exercises......Page 107
5.1 Introduction to Equality Constrained Optimization......Page 108
5.2 Newton's Method with the KKT Matrix......Page 109
5.2.1 Nonsingularity of KKT Matrix......Page 111
5.2.2 Inertia of KKT Matrix......Page 112
5.3.1 Full-Space Newton Steps......Page 113
5.3.2 Reduced-Space Newton Steps......Page 116
5.4 Quasi-Newton Methods......Page 119
5.4.1 A Quasi-Newton Full-Space Method......Page 120
5.4.2 A Quasi-Newton Reduced-Space Method......Page 122
5.5.1 Concepts of Merit Functions......Page 126
5.5.2 Filter Method Concepts......Page 129
5.5.3 Filter versus Merit Function Strategies......Page 130
5.6 Line Search Methods......Page 131
5.6.1 Line Search with Merit Functions......Page 132
5.6.2 Line Search Filter Method......Page 136
5.7 Trust Region Methods......Page 139
5.7.1 Trust Regions with Merit Functions......Page 140
5.7.2 Filter Trust Region Methods......Page 143
5.8.1 The Maratos Effect......Page 145
5.9 Summary and Conclusions......Page 147
5.11 Exercises......Page 148
6.1 Constrained NLP Formulations......Page 150
6.2 SQP Methods......Page 152
6.2.1 The Basic, Full-Space SQP Algorithm......Page 154
6.2.2 Large-Scale SQP......Page 161
6.2.3 Extensions of SQP Methods......Page 165
6.3 Interior Point Methods......Page 168
6.3.1 Solution of the Primal-Dual Equations......Page 171
6.3.2 A Line Search Filter Method......Page 172
6.3.3 Globalization with Trust Region Methods......Page 175
6.4 Nested Strategies......Page 177
6.4.1 Gradient Projection Methods for Bound Constrained Problems......Page 181
6.4.2 Linearly Constrained Augmented Lagrangian......Page 184
6.5 Nonlinear Programming Codes......Page 185
6.5.1 SQP Codes......Page 186
6.5.2 Interior Point NLP Codes......Page 187
6.5.4 Performance Trends for NLP Codes......Page 188
6.6 Summary and Conclusions......Page 192
6.7 Notes for Further Reading......Page 193
6.8 Exercises......Page 195
7.1 Introduction......Page 198
7.2 Optimization of Process Flowsheets......Page 200
7.2.1 Importance of Accurate Derivatives......Page 205
7.2.2 Ammonia Process Optimization......Page 208
7.3 Equation-Oriented Formulation of Optimization Models......Page 210
7.3.1 Reformulation of the Williams-Otto Optimization Problem......Page 213
7.4 Real-Time Optimization......Page 217
7.4.1 Equation-Oriented RTO Models......Page 218
7.4.2 Case Study of Hydrocracker Fractionation Plant......Page 220
7.5 Equation-Oriented Models with Many Degrees of Freedom......Page 223
7.7 Exercises......Page 226
8.1 Introduction......Page 230
8.2 Dynamic Systems and Optimization Problems......Page 231
8.3 Optimality Conditions for Optimal Control Problems......Page 237
8.3.1 Optimal Control without Inequalities......Page 240
8.3.2 Optimal Control with Inequality Constraints......Page 242
8.4.1 Treatment of Equality Path Constraints......Page 249
8.4.2 Treatment of State Path Inequalities......Page 254
8.5 Singular Control Problems......Page 256
8.6 Numerical Methods Based on NLP Solvers......Page 260
8.7 Summary and Conclusions......Page 263
8.8 Notes for Further Reading......Page 264
8.9 Exercises......Page 265
9.1 Introduction......Page 268
9.2 DAE Solvers for Initial Value Problems......Page 270
9.2.1 Runge-Kutta Methods......Page 272
9.2.2 Linear Multistep Methods......Page 273
9.2.3 Extension of Methods to DAEs......Page 276
9.3 Sensitivity Strategies for Dynamic Optimization......Page 277
9.3.1 Direct Sensitivity Calculations......Page 278
9.3.2 Adjoint Sensitivity Calculations......Page 279
9.3.3 Evolution to Optimal Control Problems......Page 282
9.4 Multiple Shooting......Page 288
9.4.1 Dichotomy of Boundary Value Problems......Page 290
9.5 Dynamic Optimization Case Study......Page 293
9.6 Summary and Conclusions......Page 299
9.7 Notes for Further Reading......Page 300
9.8 Exercises......Page 301
10.1 Introduction......Page 304
10.2 Derivation of Collocation Methods......Page 305
10.2.1 Polynomial Representation for ODE Solutions......Page 306
10.2.2 Collocation with Orthogonal Polynomials......Page 307
10.3 NLP Formulations and Solution......Page 312
10.3.1 Treatment of Finite Elements......Page 313
10.3.2 Treatment of Unstable Dynamic Systems......Page 316
10.3.3 Large-Scale NLP Solution Strategies......Page 319
10.3.4 Parameter Estimation for Low-Density Polyethylene Reactors......Page 321
10.4 Convergence Properties of Simultaneous Approach......Page 326
10.4.1 Optimality with Gauss-Legendre Collocation......Page 329
10.4.2 Optimality with Radau Collocation......Page 330
10.4.3 Convergence Orders for NLP Solutions......Page 331
10.4.4 Singular Optimal Controls......Page 332
10.4.5 High-Index Inequality Path Constraints......Page 334
10.6 Notes for Further Reading......Page 339
10.7 Exercises......Page 340
11.1 Introduction......Page 342
11.2 MPCC Properties and Formulations......Page 344
11.2.1 Solution Strategies......Page 348
11.2.2 Comparison of MPCC Formulations......Page 350
11.3 Complementary Models for Process Optimization......Page 353
11.4.1 Multicomponent Column Optimization with Phase Changes......Page 360
11.4.2 Tray Optimization......Page 362
11.5 Optimization of Hybrid Dynamic Systems......Page 366
11.6.1 Reformulation of a Differential Inclusion......Page 369
11.6.2 Cascading Tank Problem......Page 373
11.7 Summary and Conclusions......Page 376
11.8 Notes for Further Reading......Page 377
11.9 Exercises......Page 378
Bibliography......Page 380
Index......Page 408