by Chung Shun Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 158-170). / Abstracts in English and Chinese. / List of Figures --- p.x / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background and Historical Development --- p.1 / Chapter 1.2 --- Practical Problems --- p.3 / Chapter 1.2.1 --- Statistics --- p.3 / Chapter 1.2.2 --- Aerodynamics --- p.4 / Chapter 1.2.3 --- Factory Allocation Problem --- p.5 / Chapter 1.2.4 --- Parameter Problem --- p.5 / Chapter 1.2.5 --- Chemical Engineering --- p.5 / Chapter 1.2.6 --- Operational Research --- p.6 / Chapter 1.2.7 --- Economics --- p.6 / Chapter 1.3 --- Mathematical Models for Optimization Problems --- p.6 / Chapter 1.4 --- Unconstrained Optimization Techniques --- p.8 / Chapter 1.4.1 --- Direct Method - Differential Calculus --- p.8 / Chapter 1.4.2 --- Iterative Methods --- p.10 / Chapter 1.5 --- Main Objectives of the Thesis --- p.11 / Chapter 2 --- Basic Concepts in Optimizations of Smooth Func- tions --- p.14 / Chapter 2.1 --- Notation --- p.14 / Chapter 2.2 --- Different Types of Minimizer --- p.16 / Chapter 2.3 --- Necessary and Sufficient Conditions for Optimality --- p.18 / Chapter 2.4 --- Quadratic Functions --- p.22 / Chapter 2.5 --- Convex Functions --- p.24 / Chapter 2.6 --- "Existence, Uniqueness and Stability of a Minimum" --- p.29 / Chapter 2.6.1 --- Existence of a Minimum --- p.29 / Chapter 2.6.2 --- Uniqueness of a Minimum --- p.30 / Chapter 2.6.3 --- Stability of a Minimum --- p.31 / Chapter 2.7 --- Types of Convergence --- p.34 / Chapter 2.8 --- Minimization of Functionals --- p.35 / Chapter 3 --- Steepest Descent Method --- p.37 / Chapter 3.1 --- Background --- p.37 / Chapter 3.2 --- Line Search Method and the Armijo Rule --- p.39 / Chapter 3.3 --- Steplength Control with Polynomial Models --- p.43 / Chapter 3.3.1 --- Quadratic Polynomial Model --- p.43 / Chapter 3.3.2 --- Safeguarding --- p.45 / Chapter 3.3.3 --- Cubic Polynomial Model --- p.46 / Chapter 3.3.4 --- General Line Search Strategy --- p.49 / Chapter 3.3.5 --- Algorithm of Steepest Descent Method --- p.51 / Chapter 3.4 --- Advantages of the Armijo Rule --- p.54 / Chapter 3.5 --- Convergence Analysis --- p.56 / Chapter 4 --- Iterative Methods Using Second Derivatives --- p.63 / Chapter 4.1 --- Background --- p.63 / Chapter 4.2 --- Newton's Method --- p.64 / Chapter 4.2.1 --- Basic Concepts --- p.64 / Chapter 4.2.2 --- Convergence Analysis of Newton's Method --- p.65 / Chapter 4.2.3 --- Newton's Method with Steplength --- p.69 / Chapter 4.2.4 --- Convergence Analysis of Newton's Method with Step-length --- p.70 / Chapter 4.3 --- Greenstadt's Method --- p.72 / Chapter 4.4 --- Marquardt-Levenberg Method --- p.74 / Chapter 4.5 --- Fiacco and McComick Method --- p.76 / Chapter 4.6 --- Matthews and Davies Method --- p.79 / Chapter 4.7 --- Numerically Stable Modified Newton's Method --- p.80 / Chapter 4.8 --- The Role of the Second Derivative Methods --- p.89 / Chapter 5 --- Multi-step Methods --- p.92 / Chapter 5.1 --- Background --- p.93 / Chapter 5.2 --- Heavy Ball Method --- p.94 / Chapter 5.3 --- Conjugate Gradient Method --- p.99 / Chapter 5.3.1 --- Some Types of Conjugate Gradient Method --- p.99 / Chapter 5.3.2 --- Convergence Analysis of Conjugate Gradient Method --- p.108 / Chapter 5.4 --- Methods of Variable Metric and Methods of Conju- gate Directions --- p.111 / Chapter 5.5 --- Other Approaches for Constructing the First-order Methods --- p.116 / Chapter 6 --- Quasi-Newton Methods --- p.121 / Chapter 6.1 --- Disadvantages of Newton's Method --- p.122 / Chapter 6.2 --- General Idea of Quasi-Newton Method --- p.124 / Chapter 6.2.1 --- Quasi-Newton Methods --- p.124 / Chapter 6.2.2 --- Convergence of Quasi-Newton Methods --- p.129 / Chapter 6.3 --- Properties of Quasi-Newton Methods --- p.131 / Chapter 6.4 --- Some Particular Algorithms for Quasi-Newton Methods --- p.137 / Chapter 6.4.1 --- Single-Rank Algorithms --- p.137 / Chapter 6.4.2 --- Double-Rank Algorithms --- p.144 / Chapter 6.4.3 --- Other Applications --- p.149 / Chapter 6.5 --- Conclusion --- p.152 / Chapter 7 --- Choice of Methods in Optimization Problems --- p.154 / Chapter 7.1 --- Choice of Methods --- p.154 / Chapter 7.2 --- Conclusion --- p.157 / Bibliography --- p.158
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_323794 |
| Date | January 2002 |
| Contributors | Chung, Shun Shing., Chinese University of Hong Kong Graduate School. Division of Mathematics. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography |
| Format | print, x, 170 leaves : ill. ; 30 cm. |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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