Short Bios of the Authors xvii
Preface xxi
1 Introduction 1
1.1 Concepts of Optimization 1
1.1.1 Convex Analysis 1
1.1.1.1 Properties of Convex Sets 2
1.1.1.2 Properties of Convex Functions 2
1.1.2 Optimality Conditions 3
1.1.2.1 Karush–Kuhn–Tucker Necessary Optimality Conditions 5
1.1.2.2 Karun–Kush–Tucker First-Order Sufficient Optimality Conditions 5
1.1.3 Interpretation of Lagrange Multipliers 6
1.2 Concepts of Multi-parametric Programming 6
1.2.1 Basic Sensitivity Theorem 6
1.3 Polytopes 9
1.3.1 Approaches for the Removal of Redundant Constraints 11
1.3.1.1 Lower-Upper Bound Classification 12
1.3.1.2 Solution of Linear Programming Problem 13
1.3.2 Projections 13
1.3.3 Modeling of the Union of Polytopes 14
1.4 Organization of the Book 16
References 16
Part I Multi-parametric Optimization 19
2 Multi-parametric Linear Programming 21
2.1 Solution Properties 22
2.1.1 Local Properties 23
2.1.2 Global Properties 25
2.2 Degeneracy 28
2.2.1 Primal Degeneracy 29
2.2.2 Dual Degeneracy 30
2.2.3 Connections Between Degeneracy and Optimality Conditions 31
2.3 Critical Region Definition 32
2.4 An Example: Chicago to Topeka 33
2.4.1 The Deterministic Solution 34
2.4.2 Considering Demand Uncertainty 35
2.4.3 Interpretation of the Results 36
2.5 Literature Review 38
References 39
3 Multi-Parametric Quadratic Programming 45
3.1 Calculation of the Parametric Solution 47
3.1.1 Solution via the Basic Sensitivity Theorem 47
3.1.2 Solution via the Parametric Solution of the KKT Conditions 48
3.2 Solution Properties 49
3.2.1 Local Properties 49
3.2.2 Global Properties 50
3.2.3 Structural Analysis of the Parametric Solution 52
3.3 Chicago to Topeka with Quadratic Distance Cost 55
3.3.1 Interpretation of the Results 56
3.4 Literature Review 61
References 63
4 Solution Strategies for mp-LP and mp-QP Problems 67
4.1 General Overview 68
4.2 The Geometrical Approach 70
4.2.1 Define A Starting Point ?0 70
4.2.2 Fix ?0 in Problem (4.1), and Solve the Resulting QP 71
4.2.3 Identify The Active Set for The Solution of The QP Problem 72
4.2.4 Move Outside the Found Critical Region and Explore the Parameter Space 72
4.3 The Combinatorial Approach 75
4.3.1 Pruning Criterion 76
4.4 The Connected-Graph Approach 78
4.5 Discussion 81
4.6 Literature Review 83
References 85
5 Multi-parametric Mixed-integer Linear Programming 89
5.1 Solution Properties 90
5.1.1 From mp-LP to mp-MILP Problems 90
5.1.2 The Properties 91
5.2 Comparing the Solutions from Different mp-LP Problems 92
5.2.1 Identification of Overlapping Critical Regions 93
5.2.2 Performing the Comparison 95
5.2.3 Constraint Reversal for Coverage of Parameter Space 95
5.3 Multi-parametric Integer Linear Programming 96
5.4 Chicago to Topeka Featuring a Purchase Decision 99
5.4.1 Interpretation of the Results 99
5.5 Literature Review 102
References 103
6 Multi-parametric Mixed-integer Quadratic Programming 107
6.1 Solution Properties 109
6.1.1 From mp-QP to mp-MIQP Problems 109
6.1.2 The Properties 109
6.2 Comparing the Solutions from Different mp-QP Problems 110
6.2.1 Identification of overlapping critical regions 112
6.2.2 Performing the Comparison 112
6.3 Envelope of Solutions 113
6.4 Chicago to Topeka Featuring Quadratic Cost and A Purchase Decision 114
6.4.1 Interpretation of the Results 115
6.5 Literature Review 119
References 121
7 Solution Strategies for mp-MILP and mp-MIQP Problems 125
7.1 General Framework 126
7.2 Global Optimization 127
7.2.1 Introducing Suboptimality 129
7.3 Branch-and-Bound 130
7.4 Exhaustive Enumeration 133
7.5 The Comparison Procedure 134
7.5.1 Affine Comparison 135
7.5.2 Exact Comparison 137
7.6 Discussion 138
7.6.1 Integer Handling 138
7.6.2 Comparison Procedure 141
7.7 Literature Review 142
References 144
8 Solving Multi-parametric Programming Problems Using MATLAB® 147
8.1 An Overview over the Functionalities of POP 148
8.2 Problem Solution 148
8.2.1 Solution of mp-QP Problems 148
8.2.2 Solution of mp-MIQP Problems 148
8.2.3 Requirements and Validation 149
8.2.4 Handling of Equality Constraints 149
8.2.5 Solving Problem (7.2) 149
8.3 Problem Generation 150
8.4 Problem Library 151
8.4.1 Merits and Shortcomings of The Problem Library 152
8.5 Graphical User Interface (GUI) 153
8.6 Computational Performance for Test Sets 154
8.6.1 Continuous Problems 154
8.6.2 Mixed-integer Problems 154
8.7 Discussion 156
Acknowledgments 162
References 162
9 Other Developments in Multi-parametric Optimization 165
9.1 Multi-parametric Nonlinear Programming 165
9.1.1 The Convex Case 166
9.1.2 The Non-convex Case 167
9.2 Dynamic Programming via Multi-parametric Programming 167
9.2.1 Direct and Indirect Approaches 169
9.3 Multi-parametric Linear Complementarity Problem 170
9.4 Inverse Multi-parametric Programming 171
9.5 Bilevel Programming Using Multi-parametric Programming 172
9.6 Multi-parametric Multi-objective Optimization 173
References 174
Part II Multi-parametric Model Predictive Control 187
10 Multi-parametric/Explicit Model Predictive Control 189
10.1 Introduction 189
10.2 From Transfer Functions to Discrete Time State-Space Models 191
10.3 From Discrete Time State-Space Models to Multi-parametric Programming 195
10.4 Explicit LQR – An Example of mp-MPC 200
10.4.1 Problem Formulation and Solution 200
10.4.2 Results and Validation 202
10.5 Size of the Solution and Online Computational Effort 206
References 207
11 Extensions to Other Classes of Problems 211
11.1 Hybrid Explicit MPC 211
11.1.1 Explicit Hybrid MPC – An Example of mp-MPC 213
11.1.2 Results and Validation 215
11.2 Disturbance Rejection 219
11.2.1 Explicit Disturbance Rejection – An Example of mp-MPC 220
11.2.2 Results and Validation 222
11.3 Reference Trajectory Tracking 222
11.3.1 Reference Tracking to LQR Reformulation 227
11.3.2 Explicit Reference Tracking – An Example of mp-MPC 230
11.3.3 Results and Validation 232
11.4 Moving Horizon Estimation 232
11.4.1 Multi-parametric Moving Horizon Estimation 232
11.4.1.1 Current State 237
11.4.1.2 Recent Developments 237
11.4.1.3 Future Outlook 238
11.5 Other Developments in Explicit MPC 239
References 240
12 PAROC: PARametric Optimization and Control 243
12.1 Introduction 243
12.2 The PAROC Framework 246
12.2.1 “High Fidelity” Modeling and Analysis 247
12.2.2 Model Approximation 247
12.2.2.1 Model Approximation Algorithms: A User Perspective Within the PAROC Framework 247
12.2.3 Multi-parametric Programming 257
12.2.4 Multi-parametric Moving Horizon Policies 259
12.2.5 Software Implementation and Closed-LoopValidation 259
12.2.5.1 Multi-parametric Programming Software 259
12.2.5.2 Integration of PAROC in gPROMS® ModelBuilder 260
12.3 Case Study: Distillation Column 261
12.3.1 “High Fidelity” Modeling 262
12.3.2 Model Approximation 264
12.3.3 Multi-parametric Programming, Control, and Estimation 265
12.3.4 Closed-Loop Validation 267
12.3.5 Conclusion 268
12.4 Case Study: Simple Buffer Tank 269
12.5 The Tank Example 269
12.5.1 “High Fidelity” Dynamic Modeling 269
12.5.2 Model Approximation 270
12.5.3 Design of the Multi-parametric Model Predictive Controller 271
12.5.4 Closed-Loop Validation 272
12.5.5 Conclusion 273
12.6 Concluding Remarks 273
References 273
A Appendix for the mp-MPC Chapter 10 281
B Appendix for the mp-MPC Chapter 11 285
B.1 Matrices for the mp-QP Problem Corresponding to the
Example of Section 11.3.2 285
Index 291
