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Optimization : 100 examples /

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"This book is devoted to the analysis of scenarios for which the use of well-known optimization methods encounter certain difficulties. Analysing such examples allows a deeper understanding of the features of these optimization methods, including the limits of their applicability. In this way,...

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Bibliographic Details
Main Author: Serovajsky, Simon (Author)
Corporate Author: Taylor & Francis
Format: eBook
Language:English
Published: Boca Raton : C&H/CRC Press, 2024.
Edition:First edition.
Subjects:
Mathematical optimization.
Control theory.
Optimisation mathématique.
Théorie de la commande.
MATHEMATICS / Applied
Electronic books.
Online Access:Connect to the full text of this electronic book
Table of Contents:
  • Cover
  • Half Title
  • Title Page
  • Copyright Page
  • Dedication
  • Contents
  • Preface
  • PART I: MINIMIZATION OF FUNCTIONS OF ONE VARIABLE
  • CHAPTER 1: Fermat theorem
  • 1.1. LECTURE
  • 1.1.1. Fermat theorem
  • 1.1.2. Non-uniqueness of the solution of the stationary condition
  • 1.1.3. Absence of function minimum points
  • 1.1.4. Inapplicability of Fermat theorem
  • 1.2. APPENDIX
  • 1.2.1. Existence of function minimum
  • 1.2.2. Uniqueness of function minimum
  • 1.2.3. Tikhonov well-posedness of problems
  • 1.2.4. Sufficient conditions of function minimum
  • 1.2.5. Minimization of non-smooth functions
  • 1.2.6. Minimization of functions of many variables
  • CHAPTER 2: Additions
  • 2.1. LECTURE
  • 2.1.1. Variational inequality
  • 2.1.2. Dependence of the solution on parameters
  • 2.1.3. Approximate solving of the stationary condition
  • 2.2. APPENDIX
  • 2.2.1. Sufficiency of the minimum condition in the form of variational inequalities
  • 2.2.2. Lagrange multiplier method
  • 2.2.3. Penalty method
  • 2.2.4. Gradient methods
  • 2.2.5. Hadamard well-posedness of extremum problems
  • 2.2.6. Approximate solutions to the function minimization problem
  • 2.2.7. Minimization of functions of many variables
  • PART II: OPTIMAL CONTROL PROBLEMS FOR SYSTEMS WITH A FREE FINITE STATE
  • CHAPTER 3: Maximum principle
  • 3.1. LECTURE
  • 3.1.1. Statement of the optimal control problem
  • 3.1.2. Maximum principle
  • 3.1.3. Analytical solving of an optimal control problem
  • 3.1.4. Approximate solving of an optimal control problem
  • 3.2. APPENDIX
  • 3.2.1. Existence of function minimum
  • 3.2.2. Elimination method
  • 3.2.3. Decoupling method
  • 3.2.4. Algorithm Convergence for Example 3.3
  • 3.2.5. Vector optimal control problem
  • CHAPTER 4: Alternative methods
  • 4.1. LECTURE
  • 4.1.1. Iterative methods for solving an optimization problem
  • 4.1.2. Variational inequality
  • 4.1.3. Penalty method
  • 4.1.4. Bellman equation
  • 4.2. APPENDIX
  • 4.2.1. Problem with a non-smooth functional
  • 4.2.2. Non-equivalence of the variational inequality and maximum condition
  • 4.2.3. Penalty method in the optimal control problem with constraints
  • 4.2.4. Optimal control of a singular system
  • 4.2.5. Optimal control of a singular system with constraints
  • 4.2.6. Justification of the sufficient optimality condition
  • 4.2.7. Relationship between dynamic programming and the maximum principle
  • 4.2.8. Applicability of Bellman optimality principle
  • CHAPTER 5: Uniqueness and sufficiency
  • 5.1. LECTURE
  • 5.1.1. Problem statement
  • 5.1.2. Maximum principle
  • 5.1.3. Analysis of optimality conditions
  • 5.1.4. Uniqueness of the optimal control
  • 5.1.5. Completion of the analysis of optimality conditions
  • 5.2. APPENDIX
  • 5.2.1. Invariance of the solution under sign change
  • 5.2.2. Sufficiency of the maximum principle
  • 5.2.3. Properties of non-optimal solutions of the maximum principle