Optimization : insights and applications /

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Bibliographic Details
Author / Creator:	Brinkhuis, Jan.
Imprint:	Princeton, N.J. : Princeton University Press, c2005.
Description:	xxiv, 658 p. : ill. ; 24 cm.
Language:	English
Series:	Princeton series in applied mathematics
Subject:	Econometrics -- Mathematical models. Econometrics -- Mathematical models.
Format:	Print Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/5921237

Hidden Bibliographic Details
Other authors / contributors:	Tikhomirov, V. M. (Vladimir Mikhaĭlovich), 1934-
ISBN:	0691102872 (alk. paper)
Notes:	Includes bibliographical references (p. [645]-650) and index.

Table of Contents:

Preface
0.1. Optimization: insights and applications
0.2. Lunch, dinner, and dessert
0.3. For whom is this book meant?
0.4. What is in this book?
0.5. Special features
Necessary Conditions: What Is the Point?
Chapter 1. Fermat: One Variable without Constraints
1.0. Summary
1.1. Introduction
1.2. The derivative for one variable
1.3. Main result: Fermat theorem for one variable
1.4. Applications to concrete problems
1.5. Discussion and comments
1.6. Exercises
Chapter 2. Fermat: Two or More Variables without Constraints
2.0. Summary
2.1. Introduction
2.2. The derivative for two or more variables
2.3. Main result: Fermat theorem for two or more variables
2.4. Applications to concrete problems
2.5. Discussion and comments
2.6. Exercises
Chapter 3. Lagrange: Equality Constraints
3.0. Summary
3.1. Introduction
3.2. Main result: Lagrange multiplier rule
3.3. Applications to concrete problems
3.4. Proof of the Lagrange multiplier rule
3.5. Discussion and comments
3.6. Exercises
Chapter 4. Inequality Constraints and Convexity
4.0. Summary
4.1. Introduction
4.2. Main result: Karush-Kuhn-Tucker theorem
4.3. Applications to concrete problems
4.4. Proof of the Karush-Kuhn-Tucker theorem
4.5. Discussion and comments
4.6. Exercises
Chapter 5. Second Order Conditions
5.0. Summary
5.1. Introduction
5.2. Main result: second order conditions
5.3. Applications to concrete problems
5.4. Discussion and comments
5.5. Exercises
Chapter 6. Basic Algorithms
6.0. Summary
6.1. Introduction
6.2. Nonlinear optimization is difficult
6.3. Main methods of linear optimization
6.4. Line search
6.5. Direction of descent
6.6. Quality of approximation
6.7. Center of gravity method
6.8. Ellipsoid method
6.9. Interior point methods
Chapter 7. Advanced Algorithms
7.1. Introduction
7.2. Conjugate gradient method
7.3. Self-concordant barrier methods
Chapter 8. Economic Applications
8.1. Why you should not sell your house to the highest bidder
8.2. Optimal speed of ships and the cube law
8.3. Optimal discounts on airline tickets with a Saturday stayover
8.4. Prediction of flows of cargo
8.5. Nash bargaining
8.6. Arbitrage-free bounds for prices
8.7. Fair price for options: formula of Black and Scholes
8.8. Absence of arbitrage and existence of a martingale
8.9. How to take a penalty kick, and the minimax theorem
8.10. The best lunch and the second welfare theorem
Chapter 9. Mathematical Applications
9.1. Fun and the quest for the essence
9.2. Optimization approach to matrices
9.3. How to prove results on linear inequalities
9.4. The problem of Apollonius
9.5. Minimization of a quadratic function: Sylvester's criterion and Gram's formula
9.6. Polynomials of least deviation
9.7. Bernstein inequality
Chapter 10. Mixed Smooth-Convex Problems
10.1. Introduction
10.2. Constraints given by inclusion in a cone
10.3. Main result: necessary conditions for mixed smooth-convex problems
10.4. Proof of the necessary conditions
10.5. Discussion and comments
Chapter 11. Dynamic Programming in Discrete Time
11.0. Summary
11.1. Introduction
11.2. Main result: Hamilton-Jacobi-Bellman equation
11.3. Applications to concrete problems
11.4. Exercises
Chapter 12. Dynamic Optimization in Continuous Time
12.1. Introduction
12.2. Main results: necessary conditions of Euler, Lagrange, Pontryagin, and Bellman
12.3. Applications to concrete problems
12.4. Discussion and comments
Appendix A. On Linear Algebra: Vector and Matrix Calculus
A.1. Introduction
A.2. Zero-sweeping or Gaussian elimination, and a formula for the dimension of the solution set
A.3. Cramer's rule
A.4. Solution using the inverse matrix
A.5. Symmetric matrices
A.6. Matrices of maximal rank
A.7. Vector notation
A.8. Coordinate free approach to vectors and matrices
Appendix B. On Real Analysis
B.1. Completeness of the real numbers
B.2. Calculus of differentiation
B.3. Convexity
B.4. Differentiation and integration
Appendix C. The Weierstrass Theorem on Existence of Global Solutions
C.1. On the use of the Weierstrass theorem
C.2. Derivation of the Weierstrass theorem
Appendix D. Crash Course on Problem Solving
D.1. One variable without constraints
D.2. Several variables without constraints
D.3. Several variables under equality constraints
D.4. Inequality constraints and convexity
Appendix E. Crash Course on Optimization Theory: Geometrical Style
E.1. The main points
E.2. Unconstrained problems
E.3. Convex problems
E.4. Equality constraints
E.5. Inequality constraints
E.6. Transition to infinitely many variables
Appendix F. Crash Course on Optimization Theory: Analytical Style
F.1. Problem types
F.2. Definitions of differentiability
F.3. Main theorems of differential and convex calculus
F.4. Conditions that are necessary and/or sufficient
F.5. Proofs
Appendix G. Conditions of Extremum from Fermat to Pontryagin
G.1. Necessary first order conditions from Fermat to Pontryagin
G.2. Conditions of extremum of the second order
Appendix H. Solutions of Exercises of Chapters 1-4
Bibliography
Index

Optimization : insights and applications /

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