Optimization
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Jan Brinkhuis
and Vladimir Tikhomirov
About this book
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.
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Frontmatter
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Contents
vii -
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Preface
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Necessary Conditions: What Is the Point?
1 -
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Chapter 1. Fermat: One Variable without Constraints
3 -
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Chapter 2. Fermat: Two or More Variables without Constraints
85 -
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Chapter 3. Lagrange: Equality Constraints
135 -
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Chapter 4. Inequality Constraints and Convexity
199 -
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Chapter 5. Second Order Conditions
261 -
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Chapter 6. Basic Algorithms
273 -
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Chapter 7. Advanced Algorithms
325 -
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Chapter 8. Economic Applications
363 -
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Chapter 9. Mathematical Applications
391 -
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Chapter 10. Mixed Smooth-Convex Problems
417 -
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Chapter 11. Dynamic Programming in Discrete Time
441 -
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Chapter 12. Dynamic Optimization in Continuous Time
475 -
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Appendix A. On Linear Algebra: Vector and Matrix Calculus
503 -
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Appendix B. On Real Analysis
519 -
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Appendix C. The Weierstrass Theorem on Existence of Global Solutions
537 -
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Appendix D. Crash Course on Problem Solving
547 -
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Appendix E. Crash Course on Optimization Theory: Geometrical Style
553 -
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Appendix F. Crash Course on Optimization Theory: Analytical Style
561 -
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Appendix G. Conditions of Extremum from Fermat to Pontryagin
583 -
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Appendix H. Solutions of Exercises of Chapters 1–4
601 -
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Bibliography
645 -
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Index
651