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Efstratios N. Pistikopoulos
Multi-parametric Optimization and Control
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Вернуться на страницу книги Multi-parametric Optimization and Control
Оглавление
Страница 1
Table of Contents
List of Tables
List of Illustrations
Guide
Pages
Wiley Series in
Multi‐parametric Optimization and Control
Страница 9
Страница 10
Страница 11
Preface
Страница 13
1 Introduction
1.1 Concepts of Optimization 1.1.1 Convex Analysis
Definition 1.1 (Line)
Definition 1.2 (Line Segment)
Definition 1.3 (Convex Set)
1.1.1.1 Properties of Convex Sets
Definition 1.4 (Convex Function)
Definition 1.5 (Concave Function)
1.1.1.2 Properties of Convex Functions
1.1.2 Optimality Conditions
Definition 1.6 (Local Minimum)
Definition 1.7 (Global Minimum)
Definition 1.8 (Active Constraints)
Remark 1.1
1.1.2.1 Karush–Kuhn–Tucker Necessary Optimality Conditions
1.1.2.2 Karun–Kush–Tucker First‐Order Sufficient Optimality Conditions
1.1.3 Interpretation of Lagrange Multipliers
1.2 Concepts of Multi‐parametric Programming 1.2.1 Basic Sensitivity Theorem
Theorem 1.1 (Basic Sensitivity Theorem, [1])
Proof
1.3 Polytopes
Definition 1.9
Definition 1.10
1.3.1 Approaches for the Removal of Redundant Constraints
Theorem 1.2 ([3])
Remark 1.3
Remark 1.4
1.3.1.1 Lower‐Upper Bound Classification
1.3.1.2 Solution of Linear Programming Problem
Remark 1.5
1.3.2 Projections
Definition 1.11 (Projection [7])
Definition 1.12 (Hybrid Projection)
1.3.3 Modeling of the Union of Polytopes
1.4 Organization of the Book
References
Notes
2 Multi‐parametric Linear Programming
Remark 2.1
2.1 Solution Properties Remark 2.2
2.1.1 Local Properties
Remark 2.3
Lemma 2.1
Proof
Lemma 2.2
Proof
2.1.2 Global Properties
Theorem 2.1 (The Solution of mp‐LP Problems)
Proof
Definition 2.1 (mp‐LP Graph)
Theorem 2.2 (
The Connected‐graph Theorem
)
Proof
2.2 Degeneracy
2.2.1 Primal Degeneracy
2.2.2 Dual Degeneracy
Remark 2.4
2.2.3 Connections Between Degeneracy and Optimality Conditions
2.3 Critical Region Definition
2.4 An Example: Chicago to Topeka
2.4.1 The Deterministic Solution
2.4.2 Considering Demand Uncertainty
Remark 2.5
2.4.3 Interpretation of the Results
2.5 Literature Review
References
Notes
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