Numerical Optimization Techniques

ISBN-13: 9781461295303 / Angielski / Miękka / 2012 / 562 str.

Yurij G. Evtushenko; J. Stoer

Opis

The book of Professor Evtushenko describes both the theoretical foundations and the range of applications of many important methods for solving nonlinear programs. Particularly emphasized is their use for the solution of optimal control problems for ordinary differential equations. These methods were instrumented in a library of programs for an interactive system (DISO) at the Computing Center of the USSR Academy of Sciences, which can be used to solve a given complicated problem by a combination of appropriate methods in the interactive mode. Many examples show the strong as well the weak points of particular methods and illustrate the advantages gained by their combination. In fact, it is the central aim of the author to pOint out the necessity of using many techniques interactively, in order to solve more dif- ficult problems. A noteworthy feature of the book for the Western reader is the frequently unorthodox analysis of many known methods in the great tradition of Russian mathematics. J. Stoer PREFACE Optimization methods are finding ever broader application in sci- ence and engineering. Design engineers, automation and control systems specialists, physicists processing experimental data, eco- nomists, as well as operations research specialists are beginning to employ them routinely in their work. The applications have in turn furthered vigorous development of computational techniques and engendered new directions of research. Practical implementa- tion of many numerical methods of high computational complexity is now possible with the availability of high-speed large-memory digital computers.

Spis treści

Notification.- 1. An Introduction to Optimization Theory.- 1. Convex Sets and Convex Functions.- 2. Differentiability of Convex Functions.- 3. Necessary and Sufficient Conditions of a Local Extremum of Functions of Many Variables.- 4. Necessary and Sufficient Conditions for a Minimum of Functions on Sets.- 5. Properties of Minimax Problems.- 6. Conditions for a Minimum in Nonlinear Programming Problems Without Differentiability.- 7. Conditions for a Minimum in Nonlinear Programming Problems With Differentiability.- 8. Necessary Conditions for a Minimum in Optimal Control Problems.- 2. Convergence Theorems and Their Application to the Investigation of Numerical Methods.- 1. Stability of the First-Order Approximation.- 2. The Method of Lyapunov Functions.- 3. Theorems on Convergence of Iterative Processes.- 4. Convergence of Processes Generated by Multivalued Mappings.- 5. Methods for Solving Systems of Nonlinear Equations.- 6. Numerical Methods for Finding a Minimax.- 3. The Penalty-Function Method.- 1. The Exterior Penalty-Function Method.- 2. Estimation of Accuracy.- 3. The Cost-Function Parametrization Method.- 4. The Interior Penalty-Function Method.- 5. The Linearization Method.- 4. Numerical Methods for Solving Nonlinear Programming Problems Using Modified Lagrangians.- 1. The Simplest Modification of the Lagrangian.- 2. Modified Lagrangians.- 3. Proof of Convergence for the Simple Iteration Method.- 4. Solution of Convex Programming Problems.- 5. Reduction to a Maximin Problem.- 6. Reduction to a Minimax Problem.- 5. Relaxation Methods for Solving Nonlinear Programming Problems.- 1. Application of the Reduced Gradient Method to Solving Problems With Equality-Type Constraints.- 2. A Generalization of the Reduced Gradient Method.- 3. A Discrete Version of the Reduced Gradient Method.- 4. The Conditional Gradient Method.- 5. The Gradient Projection Method.- 6. Numerical Methods for Solving Optimal Control Problems.- 1. Basic Computational Formulas.- 2. Necessary and Sufficient Conditions for A Minimum.- 3. Numerical Methods Based on the Reduction to Nonlinear Programming Problems.- 4. Discrete Minimum Principles.- 5. Numerical Methods Based on Discrete Minimum Principle.- 6. Some Generalizations.- 7. Examples of Numerical Computations.- 8. An Application to Differential Games.- 7. Search for Global Solutions.- 1. The General Notion of Coverings.- 2. Covering a Parallelepiped.- 3. Solution of Nonlinear Programming Problems.- 4. Solution of Systems of Algebraic Equations.- 5. Solution of Minimax Problems.- 6. Solution of Multicriteria Problems.- Appendix I. Differentiability.- Appendix II. Some Properties of Matrices.- Appendix III. Some Properties of Mappings.- Notes and Comments.- References.- List of Forthcoming Publications.- transliteration Table.