Optimization : 100 examples /
Simon Serovajsky.
- First edition.
- 1 online resource.
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
"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, the book seeks to stimulate further development and understanding of the theory of optimal control. The study of the presented examples makes it possible to more effectively diagnose problems that arise in the practical solution of optimal control problems, and to find ways to overcome the difficulties that have arisen"--