An Introduction to Nonlinear Optimization Theory

✍ Scribed by Marius Durea; Radu Strugariu

• Publisher: De Gruyter Open Poland
• Year: 2014
• Tongue: English
• Leaves: 328
• Category: Library

✦ Synopsis

The goal of this book is to present the main ideas and techniques in the field of continuous smooth and nonsmooth optimization. Starting with the case of differentiable data and the classical results on constrained optimization problems, and continuing with the topic of nonsmooth objects involved in optimization theory, the book concentrates on both theoretical and practical aspects of this field. This book prepares those who are engaged in research by giving repeated insights into ideas that are subsequently dealt with and illustrated in detail.

✦ Table of Contents

Preface
1 Preliminaries
1.1 Rp Space
1.2 Limits of Functions and Continuity
1.3 Differentiability
1.4 The Riemann Integral
2 Nonlinear Analysis Fundamentals
2.1 Convex Sets and Cones
2.2 Convex Functions
2.2.1 General Results
2.2.2 Convex Functions of One Variable
2.2.3 Inequalities
2.3 Banach Fixed Point Principle
2.3.1 Contractions and Fixed Points
2.3.2 The Case of One Variable Functions
2.4 Graves Theorem
2.5 Semicontinuous Functions
3 The Study of Smooth Optimization Problems
3.1 General Optimality Conditions
3.2 Functional Restrictions
3.2.1 Fritz John Optimality Conditions
3.2.2 Karush-Kuhn-Tucker Conditions
3.2.3 Qualification Conditions
3.3 Second-order Conditions
3.4 Motivations for Scientific Computations
4 Convex Nonsmooth Optimization
4.1 Further Properties and Separation of Convex Sets
4.2 The Subdifferential of a Convex Function
4.3 Optimality Conditions
5 Lipschitz Nonsmooth Optimization
5.1 Clarke Generalized Calculus
5.1.1 Clarke Subdifferential
5.1.2 Clarke Tangent and Normal Cones
5.1.3 Optimality Conditions in Lipschitz Optimization
5.2 Mordukhovich Generalized Calculus
5.2.1 Fréchet and Mordukhovich Normal Cones
5.2.2 Fréchet and Mordukhovich Subdifferentials
5.2.3 The Extremal Principle
5.2.4 Calculus Rules
5.2.5 Optimality Conditions
6 Basic Algorithms
6.1 Algorithms for Nonlinear Equations
6.1.1 Picard's Algorithm
6.1.2 Newton's Method
6.2 Algorithms for Optimization Problems
6.2.1 The Case of Unconstrained Problems
6.2.2 The Case of Constraint Problems
6.3 Scientific Calculus Implementations
7 Exercises and Problems, and their Solutions
7.1 Analysis of Real Functions of One Variable
7.2 Nonlinear Analysis
7.3 Smooth Optimization
7.4 Nonsmooth Optimization
Bibliography
List of Notations
Index