Optimization

Insights and Applications

Jan Brinkhuis, Vladimir Tikhomirov

• Publisher: Princeton University Press
• Year: 2011
• Print ISBN: 9780691102870
• eBook ISBN: 9781400829361
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

This self-contained textbook is an informal introduction to optimization through the use of numerous illustrations and applications. The focus is on analytically solving optimization problems with a finite number of continuous variables. In addition, the authors provide introductions to classical and modern numerical methods of optimization and to dynamic optimization.


The book's overarching point is that most problems may be solved by the direct application of the theorems of Fermat, Lagrange, and Weierstrass. The authors show how the intuition for each of the theoretical results can be supported by simple geometric figures. They include numerous applications through the use of varied classical and practical problems. Even experts may find some of these applications truly surprising.


A basic mathematical knowledge is sufficient to understand the topics covered in this book. More advanced readers, even experts, will be surprised to see how all main results can be grounded on the Fermat-Lagrange theorem. The book can be used for courses on continuous optimization, from introductory to advanced, for any field for which optimization is relevant.

Table of contents

• Cover
• Title
• Copyright
• Contents
• Preface
• Necessary Conditions: What Is the Point?
• Chapter 1. Fermat: One Variable without Constraints
• Chapter 2. Fermat: Two or More Variables without Constraints
• Chapter 3. Lagrange: Equality Constraints
• Chapter 4. Inequality Constraints and Convexity
• Chapter 5. Second Order Conditions
• Chapter 6. Basic Algorithms
• Chapter 7. Advanced Algorithms
• Chapter 8. Economic Applications
• Chapter 9. Mathematical Applications
• Chapter 10. Mixed Smooth-Convex Problems
• Chapter 11. Dynamic Programming in Discrete Time
• Chapter 12. Dynamic Optimization in Continuous Time
• Appendix A. On Linear Algebra: Vector and Matrix Calculus
• Appendix B. On Real Analysis
• Appendix C. The Weierstrass Theorem on Existence of Global Solutions
• Appendix D. Crash Course on Problem Solving
• Appendix E. Crash Course on Optimization Theory: Geometrical Style
• Appendix F. Crash Course on Optimization Theory: Analytical Style
• Appendix G. Conditions of Extremum from Fermat to Pontryagin
• Appendix H. Solutions of Exercises of Chapters 1–4
• Bibliography
• Index