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Multi-parametric Optimization and Control. Edition No. 1. Wiley Series in Operations Research and Management Science

  • Book
  • 320 Pages
  • January 2021
  • John Wiley and Sons Ltd
  • ID: 5185285

Recent developments in multi-parametric optimization and control 

Multi-Parametric Optimization and Control provides comprehensive coverage of recent methodological developments for optimal model-based control through parametric optimization. It also shares real-world research applications to support deeper understanding of the material. 

Researchers and practitioners can use the book as reference. It is also suitable as a primary or a supplementary textbook. Each chapter looks at the theories related to a topic along with a relevant case study. Topic complexity increases gradually as readers progress through the chapters. The first part of the book presents an overview of the state-of-the-art multi-parametric optimization theory and algorithms in multi-parametric programming. The second examines the connection between multi-parametric programming and model-predictive control - from the linear quadratic regulator over hybrid systems to periodic systems and robust control.  

The third part of the book addresses multi-parametric optimization in process systems engineering. A step-by-step procedure is introduced for embedding the programming within the system engineering, which leads the reader into the topic of the PAROC framework and software platform. PAROC is an integrated framework and platform for the optimization and advanced model-based control of process systems. 

  • Uses case studies to illustrate real-world applications for a better understanding of the concepts presented  
  • Covers the fundamentals of optimization and model predictive control  
  • Provides information on key topics, such as the basic sensitivity theorem, linear programming, quadratic programming, mixed-integer linear programming, optimal control of continuous systems, and multi-parametric optimal control  

An appendix summarizes the history of multi-parametric optimization algorithms. It also covers the use of the parametric optimization toolbox (POP), which is comprehensive software for efficiently solving multi-parametric programming problems. 

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 𝜃0 70

4.2.2 Fix 𝜃0 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

Efstratios N. Pistikopoulos Imperial College London, Department of Chemical Engineering, London, United Kingdom. Nikolaos A. Diangelakis Richard Oberdieck