Mathematically, most of the interesting optimization problems can be formulated to optimize some objective function, subject to some equality and/or inequality constraints. This book introduces some classical and basic results of optimization theory, including nonlinear programming with Lagrange multiplier method, the Karush–Kuhn–Tucker method, Fritz John's method, problems with convex or quasi-convex constraints, and linear programming with geometric method and simplex method.
A slim book such as this which touches on major aspects of optimization theory will be very much needed for most readers. We present nonlinear programming, convex programming, and linear programming in a self-contained manner. This book is for a one-semester course for upper level undergraduate students or first/second year graduate students. It should also be useful for researchers working on many interdisciplinary areas other than optimization.
Author(s): Jiongmin Yong
Edition: 1
Publisher: World Scientific Publishing Co. Pte. Ltd.
Year: 2018
Cover
Halftitle
Title
Copyright
Dedication
Preface
Contents
1. Mathematical Preparations
1 Basics in Rn
2 Some Results of Linear Algebra
3 Functions: Limits and Continuity
4 Differentiation
2. Optimization Problems and Existence of Optimal Solutions
1 Optimization Problems
2 Some Examples of Optimization Problems
3 Existence of Optimal Solutions
4 A Constructive Method
3. Necessary and Sufficient Conditions of Optimal Solutions
1 Unconstrained Problems
2 Problems with Equality Constraints
3 Problems with Equality and Inequality Constraints
4 Problems without Regularity of the Constraints
4. Problems with Convexity and Quasi-Convexity Conditions
1 Convex Sets and Convex Functions
2 Optimization Problems under Convexity Conditions
3 Lagrange Duality
4 Quasi-Convexity and Related Optimization Problems
5. Linear Programming
1 Standard and Canonical Forms
2 Geometric Considerations and the Fundamental Theorem of Linear Programming
3 The Simplex Method
3.1 General consideration
3.2 Phase II
3.3 Phase I
3.4 Endless cycling∗
4 Sensitivity Analysis
5 Duality Theory
Bibliography
Index
