Optimization : 100 Examples

Optimization: 100 Examples is a book devoted to the analysis of scenarios for which the use of well-known optimization methods encounter certain difficulties. Analyzing 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. Features • Vast collection of examples • Simple accessible presentation • Suitable as a research reference for anyone with an interest in optimization and optimal control theory, including mathematicians and engineers • Examples differ in properties, i.e., each effect for each class of problems is illustrated by a unique example. 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 5.2.4. Variational inequality in case of insufficiency of the maximum principle 5.2.5. Elimination method 5.2.6. Modification of Example 5.1 CHAPTER 6: Singular controls 6.1. LECTURE 6.1.1. Problem statement 6.1.2. Analysis of optimality conditions 6.1.3. Singular controls 6.1.4. Existence of singular control 6.1.5. Finiteness of the set of singular controls 6.2. APPENDIX 6.2.1. Application of uniqueness and sufficiency theorems 6.2.2. Control is optimal as singular and not optimal as regular 6.2.3. Non-optimal singular controls 6.2.4. Kelley condition 6.2.5. Kopp–Moyer condition CHAPTER 7: Unsolvability of optimal control problems 7.1. LECTURE 7.1.1. Statement of the problem and its analysis 7.1.2. Unsolvability of the optimization problem 7.1.3. Existence of optimal control 7.1.4. Application of the existence theorem 7.2. APPENDIX 7.2.1. Existence of an optimal control when the set of admissible controls is un-bounded 7.2.2. Unsolvability of a problem with insufficient optimality conditions 7.2.3. Minimization of a functional on a non-convex set 7.2.4. Extension methods 7.2.5. Some features of non-linear boundary value problems CHAPTER 8: Ill-posed optimal control problems 8.1. LECTURE 8.1.1. Tikhonov well-posedness 8.1.2. Justification of Tikhonov well-posedness 8.1.3. Hadamard well-posedness 8.2. APPENDIX 8.2.1. Types of approximate solution of the problem of finding an extremum 8.2.2. Justification of Hadamard well-posedness 8.2.3. Regularization of optimal control problems PART III: OPTIMAL CONTROL PROBLEMS FOR SYSTEMS WITH A FIXED FINAL STATE CHAPTER 9: Maximum principle for systems with a fixed final state 9.1. LECTURE 9.1.1. Problem statement 9.1.2. Maximum principle 9.1.3. Example of an analytical solving to a problem 9.1.4. Approximate solving of a problem with a fixed final state 9.2. APPENDIX 9.2.1. Qualitative analysis of Example 9.1 9.2.2. Maximizing the functional from Example 9.1 9.2.3. Problem with a quadratic functional CHAPTER 10: Addition 10.1. LECTURE 10.1.1. Decoupling method 10.1.2. Variational inequality 10.1.3. Penalty method 10.2. APPENDIX 10.2.1. Applicability of Bellman optimality principle 10.2.2. Shortest Curve 10.2.3. Penalty method with functional differentiation 10.2.4. Vector optimal control problem with a fixed finite state 10.2.5. Time optimal problem CHAPTER 11: Counterexamples of optimal control problems with a fixed final state 11.1. LECTURE 11.1.1. Insufficiency of optimality conditions 11.1.2. Singular control 11.1.3. Non-uniqueness of the optimal control 11.1.4. Unsolvable optimal control problem 11.2. APPENDIX 11.2.1. Maximization of a functional with unique singular control 11.2.2. Maximization of a functional with three singular controls 11.2.3. Completion of the analysis of Example 11.4 11.2.4. Problem with infinite set of solutions 11.2.5. Problems with degeneracy of the Kelley condition CHAPTER 12: Ill-posed optimal control problems with a fixed final state 12.1. LECTURE 12.1.1. Well-posed optimal control problems with a fixed final state 12.1.2. Tikhonov ill-posed problem 12.1.3. Hadamard ill-posed problem 12.1.4. Bifurcation of extremals 12.1.5. Chafee–Infante problem 12.2. APPENDIX 12.2.1. Maximization of the functional from Example 12.1 12.2.2. Ill-posedness of the problem from Example 11.2 12.2.3. Analysis of the Chafee–Infante problem PART IV: OPTIMAL CONTROL PROBLEMS FOR SYSTEMS WITH ISOPERIMETRIC CONDITIONS CHAPTER 13: Optimization of systems with isoperimetric conditions 13.1. LECTURE 13.1.1. Optimal control problem with isoperimetric condition 13.1.2. Analytical solving of the problem with isoperimetric condition 13.1.3. Problem with isoperimetric condition and fixed final state 13.1.4. Analytical solving of a problem with an isoperimetric condition and a fixed final state 13.2. APPENDIX 13.2.1. Approximate solving of the problem with isoperimetric condition 13.2.2. Qualitative analysis of the considered examples 13.2.3. Dido problem 13.2.4. Penalty method and variational inequality 13.2.5. Vector problem with isoperimetric conditions CHAPTER 14: Absence of sufficiency and uniqueness in problems with isoperimetric conditions 14.1. LECTURE 14.1.1. Linear system of optimality conditions for a fixed final state 14.1.2. Non-linear system of optimality conditions for a fixed final state 14.2. APPENDIX 14.2.1. Non-uniqueness and insufficiency for a system with a free final state 14.2.2. System with constraints on control values 14.2.3. Existence of optimal control for Example 14.2 CHAPTER 15: Different counterexamples for optimization problems with isoperimetric conditions 15.1. LECTURE 15.1.1. Insolvability of a problem with a free final state 15.1.2. Insolvability of problem with a fixed final state 15.1.3. Singular controls 15.1.4. Ill-posed problems 15.2. APPENDIX 15.2.1. Problems with an infinite set of singular controls 15.2.2. Singular controls for problems with a fixed final state 15.2.3. Ill-posed problem with a fixed final state 15.2.4. Extremal bifurcation for problems with isoperimetric condition 15.2.5. Applicability of the Bellman principle for problems with isoperimetric conditions PART V: OPTIMAL CONTROL PROBLEMS WITH A FREE INITIAL STATE CHAPTER 16: Optimal control systems with a free initial state 16.1. LECTURE 16.1.1. Optimal control problem for a system with a free initial state 16.1.2. Maximum principle for a system with a free initial state 16.1.3. Analytical solution of the problem in the absence of control restrictions 16.1.4. Analytical solution of the problem with control constraint 16.1.5. Algorithm of solving the optimality conditions 16.2. APPENDIX 16.2.1. Qualitative analysis of examples 16.2.2. Decoupling method 16.2.3. Penalty method 16.2.4. Optimal control problem for a singular system with a free initial state CHAPTER 17: Different optimal control problems for systems with a free initial state 17.1. LECTURE 17.1.1. Non-uniqueness and non-sufficiency 17.1.2. Singular controls 17.1.3. Insolvability of an optimal control problem 17.1.4. Ill-posed optimal control problems 17.2. APPENDIX 17.2.1. Non-uniqueness and insufficiency under different controls 17.2.2. Special property of the singular control 17.2.3. Infinite set of singular controls 17.2.4. Problem with an isoperimetric condition with respect to control 17.2.5. Problem with an isoperimetric condition with respect to state List of examples Bibliography Index Optimization: 100 Examples is a book 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. Features Vast collection of examples Simple. accessible presentation Suitable as a research reference for anyone with an interest in optimization and optimal control theory, including mathematicians and engineers Examples differ in properties, i.e. each effect for each class of problems is illustrated by a unique example. Simon Serovajsky is a professor of mathematics at Al-Farabi Kazakh National University in Kazakhstan. He is the author of many books published in the area of optimization and optimal control theory, mathematical physics, mathematical modelling, philosophy and history of mathematics as well as a long list of high-quality publications in learned journals.