Multi-parametric Optimization and Control

EFSTRATIOS N. PISTIKOPOULOS is the Director of the Texas A&M Energy Institute and a TEES Eminent Professor in the Artie McFerrin Department of Chemical Engineering at Texas A&M University. He holds a Ph.D. degree from Carnegie Mellon University (1988) and was with Shell Chemicals in Amsterdam before joining Imperial. He has authored or co-authored over 500 major research publications in the areas of modelling, control and optimization of process, energy and systems engineering applications, 15 books and 2 patents. NIKOLAOS A. DIANGELAKIS is an Optimization Specialist at Octeract Ltd. He holds a PhD and MSc on Advanced Chemical Engineering from Imperial College London and was a member of the Multi-Parametric Optimization and Control group at Imperial and then Texas A&M since 2011. He is the co-author of 16 journal papers, 11 conference papers and 3 book chapters. RICHARD OBERDIECK is a Technical Account Manager at Gurobi Optimization, LLC. He obtained a bachelor and MSc degrees from ETH Zurich in Switzerland (2009-1013), before pursuing a PhD in Chemical Engineering at Imperial College London, UK, which he completed in 2017. He has published 21 papers and 2 book chapters, has an h-index of 11 and was awarded the FICO Decisions Award 2019 in Optimization, Machine Learning and AI.

• Short Bios of the Authors xviiPreface xxi1 Introduction 11.1 Concepts of Optimization 11.1.1 Convex Analysis 11.1.1.1 Properties of Convex Sets 21.1.1.2 Properties of Convex Functions 21.1.2 Optimality Conditions 31.1.2.1 Karush–Kuhn–Tucker Necessary Optimality Conditions 51.1.2.2 Karun–Kush–Tucker First-Order Sufficient Optimality Conditions 51.1.3 Interpretation of Lagrange Multipliers 61.2 Concepts of Multi-parametric Programming 61.2.1 Basic Sensitivity Theorem 61.3 Polytopes 91.3.1 Approaches for the Removal of Redundant Constraints 111.3.1.1 Lower-Upper Bound Classification 121.3.1.2 Solution of Linear Programming Problem 131.3.2 Projections 131.3.3 Modeling of the Union of Polytopes 141.4 Organization of the Book 16References 16Part I Multi-parametric Optimization 192 Multi-parametric Linear Programming 212.1 Solution Properties 222.1.1 Local Properties 232.1.2 Global Properties 252.2 Degeneracy 282.2.1 Primal Degeneracy 292.2.2 Dual Degeneracy 302.2.3 Connections Between Degeneracy and Optimality Conditions 312.3 Critical Region Definition 322.4 An Example: Chicago to Topeka 332.4.1 The Deterministic Solution 342.4.2 Considering Demand Uncertainty 352.4.3 Interpretation of the Results 362.5 Literature Review 38References 393 Multi-Parametric Quadratic Programming 453.1 Calculation of the Parametric Solution 473.1.1 Solution via the Basic Sensitivity Theorem 473.1.2 Solution via the Parametric Solution of the KKT Conditions 483.2 Solution Properties 493.2.1 Local Properties 493.2.2 Global Properties 503.2.3 Structural Analysis of the Parametric Solution 523.3 Chicago to Topeka with Quadratic Distance Cost 553.3.1 Interpretation of the Results 563.4 Literature Review 61References 634 Solution Strategies for mp-LP and mp-QP Problems 674.1 General Overview 684.2 The Geometrical Approach 704.2.1 Define A Starting Point ;;0 704.2.2 Fix ;;0 in Problem (4.1), and Solve the Resulting QP 714.2.3 Identify The Active Set for The Solution of The QP Problem 724.2.4 Move Outside the Found Critical Region and Explore the Parameter Space 724.3 The Combinatorial Approach 754.3.1 Pruning Criterion 764.4 The Connected-Graph Approach 784.5 Discussion 814.6 Literature Review 83References 855 Multi-parametric Mixed-integer Linear Programming 895.1 Solution Properties 905.1.1 From mp-LP to mp-MILP Problems 905.1.2 The Properties 915.2 Comparing the Solutions from Different mp-LP Problems 925.2.1 Identification of Overlapping Critical Regions 935.2.2 Performing the Comparison 955.2.3 Constraint Reversal for Coverage of Parameter Space 955.3 Multi-parametric Integer Linear Programming 965.4 Chicago to Topeka Featuring a Purchase Decision 995.4.1 Interpretation of the Results 995.5 Literature Review 102References 1036 Multi-parametric Mixed-integer Quadratic Programming 1076.1 Solution Properties 1096.1.1 From mp-QP to mp-MIQP Problems 1096.1.2 The Properties 1096.2 Comparing the Solutions from Different mp-QP Problems 1106.2.1 Identification of overlapping critical regions 1126.2.2 Performing the Comparison 1126.3 Envelope of Solutions 1136.4 Chicago to Topeka Featuring Quadratic Cost and A Purchase Decision 1146.4.1 Interpretation of the Results 1156.5 Literature Review 119References 1217 Solution Strategies for mp-MILP and mp-MIQP Problems 1257.1 General Framework 1267.2 Global Optimization 1277.2.1 Introducing Suboptimality 1297.3 Branch-and-Bound 1307.4 Exhaustive Enumeration 1337.5 The Comparison Procedure 1347.5.1 Affine Comparison 1357.5.2 Exact Comparison 1377.6 Discussion 1387.6.1 Integer Handling 1387.6.2 Comparison Procedure 1417.7 Literature Review 142References 1448 Solving Multi-parametric Programming Problems Using MATLAB®1478.1 An Overview over the Functionalities of POP 1488.2 Problem Solution 1488.2.1 Solution of mp-QP Problems 1488.2.2 Solution of mp-MIQP Problems 1488.2.3 Requirements and Validation 1498.2.4 Handling of Equality Constraints 1498.2.5 Solving Problem (7.2) 1498.3 Problem Generation 1508.4 Problem Library 1518.4.1 Merits and Shortcomings of The Problem Library 1528.5 Graphical User Interface (GUI) 1538.6 Computational Performance for Test Sets 1548.6.1 Continuous Problems 1548.6.2 Mixed-integer Problems 1548.7 Discussion 156Acknowledgments 162References 1629 Other Developments in Multi-parametric Optimization 1659.1 Multi-parametric Nonlinear Programming 1659.1.1 The Convex Case 1669.1.2 The Non-convex Case 1679.2 Dynamic Programming via Multi-parametric Programming 1679.2.1 Direct and Indirect Approaches 1699.3 Multi-parametric Linear Complementarity Problem 1709.4 Inverse Multi-parametric Programming 1719.5 Bilevel Programming Using Multi-parametric Programming 1729.6 Multi-parametric Multi-objective Optimization 173References 174Part II Multi-parametric Model Predictive Control 18710 Multi-parametric/Explicit Model Predictive Control 18910.1 Introduction 18910.2 From Transfer Functions to Discrete Time State-Space Models 19110.3 From Discrete Time State-Space Models to Multi-parametric Programming 19510.4 Explicit LQR – An Example of mp-MPC 20010.4.1 Problem Formulation and Solution 20010.4.2 Results and Validation 20210.5 Size of the Solution and Online Computational Effort 206References 20711 Extensions to Other Classes of Problems 21111.1 Hybrid Explicit MPC 21111.1.1 Explicit Hybrid MPC – An Example of mp-MPC 21311.1.2 Results and Validation 21511.2 Disturbance Rejection 21911.2.1 Explicit Disturbance Rejection – An Example of mp-MPC 22011.2.2 Results and Validation 22211.3 Reference Trajectory Tracking 22211.3.1 Reference Tracking to LQR Reformulation 22711.3.2 Explicit Reference Tracking – An Example of mp-MPC 23011.3.3 Results and Validation 23211.4 Moving Horizon Estimation 23211.4.1 Multi-parametric Moving Horizon Estimation 23211.4.1.1 Current State 23711.4.1.2 Recent Developments 23711.4.1.3 Future Outlook 23811.5 Other Developments in Explicit MPC 239References 24012 PAROC: PARametric Optimization and Control 24312.1 Introduction 24312.2 The PAROC Framework 24612.2.1 “High Fidelity” Modeling and Analysis 24712.2.2 Model Approximation 24712.2.2.1 Model Approximation Algorithms: A User Perspective Within the PAROC Framework 24712.2.3 Multi-parametric Programming 25712.2.4 Multi-parametric Moving Horizon Policies 25912.2.5 Software Implementation and Closed-LoopValidation 25912.2.5.1 Multi-parametric Programming Software 25912.2.5.2 Integration of PAROC in gPROMS® ModelBuilder 26012.3 Case Study: Distillation Column 26112.3.1 “High Fidelity” Modeling 26212.3.2 Model Approximation 26412.3.3 Multi-parametric Programming, Control, and Estimation 26512.3.4 Closed-Loop Validation 26712.3.5 Conclusion 26812.4 Case Study: Simple Buffer Tank 26912.5 The Tank Example 26912.5.1 “High Fidelity” Dynamic Modeling 26912.5.2 Model Approximation 27012.5.3 Design of the Multi-parametric Model Predictive Controller 27112.5.4 Closed-Loop Validation 27212.5.5 Conclusion 27312.6 Concluding Remarks 273References 273A Appendix for the mp-MPC Chapter 10 281B Appendix for the mp-MPC Chapter 11 285B.1 Matrices for the mp-QP Problem Corresponding to theExample of Section 11.3.2 285Index 291