Optimization Insights and Applications

Name: Optimization Insights and Applications
Price: 109.59 USD
Availability: InStock
ISBN: 9780691102870

ISBN-10: 0691102872

ISBN-13: 9780691102870

Edition: 2005

Authors: Jan Brinkhuis, Vladimir Tikhomirov

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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…

Book details

List price: $115.00
Copyright year: 2005
Publisher: Princeton University Press
Publication date: 9/18/2005
Binding: Hardcover
Pages: 688
Size: 6.25" wide x 9.25" long x 2.00" tall
Weight: 2.398
Language: English



Preface



Optimization: insights and applications



Lunch, dinner, and dessert



For whom is this book meant?



What is in this book?



Special features


Necessary Conditions: What Is the Point?



Fermat: One Variable without Constraints



Summary



Introduction



The derivative for one variable



Main result: Fermat theorem for one variable



Applications to concrete problems



Discussion and comments



Exercises



Fermat: Two or More Variables without Constraints



Summary



Introduction



The derivative for two or more variables



Main result: Fermat theorem for two or more variables



Applications to concrete problems



Discussion and comments



Exercises



Lagrange: Equality Constraints



Summary



Introduction



Main result: Lagrange multiplier rule



Applications to concrete problems



Proof of the Lagrange multiplier rule



Discussion and comments



Exercises



Inequality Constraints and Convexity



Summary



Introduction



Main result: Karush-Kuhn-Tucker theorem



Applications to concrete problems



Proof of the Karush-Kuhn-Tucker theorem



Discussion and comments



Exercises



Second Order Conditions



Summary



Introduction



Main result: second order conditions



Applications to concrete problems



Discussion and comments



Exercises



Basic Algorithms



Summary



Introduction



Nonlinear optimization is difficult



Main methods of linear optimization



Line search



Direction of descent



Quality of approximation



Center of gravity method



Ellipsoid method



Interior point methods



Advanced Algorithms



Introduction



Conjugate gradient method



Self-concordant barrier methods



Economic Applications



Why you should not sell your house to the highest bidder



Optimal speed of ships and the cube law



Optimal discounts on airline tickets with a Saturday stayover



Prediction of flows of cargo



Nash bargaining



Arbitrage-free bounds for prices



Fair price for options: formula of Black and Scholes



Absence of arbitrage and existence of a martingale



How to take a penalty kick, and the minimax theorem



The best lunch and the second welfare theorem



Mathematical Applications



Fun and the quest for the essence



Optimization approach to matrices



How to prove results on linear inequalities



The problem of Apollonius



Minimization of a quadratic function: Sylvester's criterion and Gram's formula



Polynomials of least deviation



Bernstein inequality



Mixed Smooth-Convex Problems



Introduction



Constraints given by inclusion in a cone



Main result: necessary conditions for mixed smooth-convex problems



Proof of the necessary conditions



Discussion and comments



Dynamic Programming in Discrete Time



Summary



Introduction



Main result: Hamilton-Jacobi-Bellman equation



Applications to concrete problems



Exercises



Dynamic Optimization in Continuous Time



Introduction



Main results: necessary conditions of Euler, Lagrange, Pontryagin, and Bellman



Applications to concrete problems



Discussion and comments



On Linear Algebra: Vector and Matrix Calculus



Introduction



Zero-sweeping or Gaussian elimination, and a formula for the dimension of the solution set



Cramer's rule



Solution using the inverse matrix



Symmetric matrices



Matrices of maximal rank



Vector notation



Coordinate free approach to vectors and matrices



On Real Analysis



Completeness of the real numbers



Calculus of differentiation



Convexity



Differentiation and integration



The Weierstrass Theorem on Existence of Global Solutions



On the use of the Weierstrass theorem



Derivation of the Weierstrass theorem



Crash Course on Problem Solving



One variable without constraints



Several variables without constraints



Several variables under equality constraints



Inequality constraints and convexity



Crash Course on Optimization Theory: Geometrical Style



The main points



Unconstrained problems



Convex problems



Equality constraints



Inequality constraints



Transition to infinitely many variables



Crash Course on Optimization Theory: Analytical Style



Problem types



Definitions of differentiability



Main theorems of differential and convex calculus



Conditions that are necessary and/or sufficient



Proofs



Conditions of Extremum from Fermat to Pontryagin



Necessary first order conditions from Fermat to Pontryagin



Conditions of extremum of the second order



Solutions of Exercises of Chapters 1-4


Bibliography


Index