Multi-Parametric Optimization and Control - Efstratios N Pistikopoulos

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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 ...