Optimization: Insights and Applications (Hardcover)

Jan Brinkhuis, Vladimir Tikhomirov

  • 出版商: Princeton University
  • 出版日期: 2005-09-18
  • 售價: $1,400
  • 貴賓價: 9.8$1,372
  • 語言: 英文
  • 頁數: 680
  • 裝訂: Hardcover
  • ISBN: 0691102872
  • ISBN-13: 9780691102870
  • 相關分類: 數值分析 Numerical-analysis
  • 下單後立即進貨 (約5~7天)
    無現貨庫存(No stock available)

買這商品的人也買了...

商品描述

Description

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

Preface xi
0.1 Optimization: insights and applications xiii
0.2 Lunch, dinner, and dessert xiv
0.3 For whom is this book meant? xvi
0.4 What is in this book? xviii
0.5 Special features xix
Necessary Conditions: What Is the Point? 1

Chapter 1. Fermat: One Variable without Constraints 3
1.0 Summary 3
1.1 Introduction 5
1.2 The derivative for one variable 6
1.3 Main result: Fermat theorem for one variable 14
1.4 Applications to concrete problems 30
1.5 Discussion and comments 43
1.6 Exercises 59

Chapter 2. Fermat: Two or More Variables without Constraints 85
2.0 Summary 85
2.1 Introduction 87
2.2 The derivative for two or more variables 87
2.3 Main result: Fermat theorem for two or more variables 96
2.4 Applications to concrete problems 101
2.5 Discussion and comments 127
2.6 Exercises 128

Chapter 3. Lagrange: Equality Constraints 135
3.0 Summary 135
3.1 Introduction 138
3.2 Main result: Lagrange multiplier rule 140
3.3 Applications to concrete problems 152
3.4 Proof of the Lagrange multiplier rule 167
3.5 Discussion and comments 181
3.6 Exercises 190

Chapter 4. Inequality Constraints and Convexity 199
4.0 Summary 199
4.1 Introduction 202
4.2 Main result: Karush-Kuhn-Tucker theorem 204
4.3 Applications to concrete problems 217
4.4 Proof of the Karush-Kuhn-Tucker theorem 229
4.5 Discussion and comments 235
4.6 Exercises 250

Chapter 5. Second Order Conditions 261
5.0 Summary 261
5.1 Introduction 262
5.2 Main result: second order conditions 262
5.3 Applications to concrete problems 267
5.4 Discussion and comments 271
5.5 Exercises 272

Chapter 6. Basic Algorithms 273
6.0 Summary 273
6.1 Introduction 275
6.2 Nonlinear optimization is difficult 278
6.3 Main methods of linear optimization 283
6.4 Line search 286
6.5 Direction of descent 299
6.6 Quality of approximation 301
6.7 Center of gravity method 304
6.8 Ellipsoid method 307
6.9 Interior point methods 316

Chapter 7. Advanced Algorithms 325
7.1 Introduction 325
7.2 Conjugate gradient method 325
7.3 Self-concordant barrier methods 335

Chapter 8. Economic Applications 363
8.1 Why you should not sell your house to the highest bidder 363
8.2 Optimal speed of ships and the cube law 366
8.3 Optimal discounts on airline tickets with a Saturday stayover 368
8.4 Prediction of ows of cargo 370
8.5 Nash bargaining 373
8.6 Arbitrage-free bounds for prices 378
8.7 Fair price for options: formula of Black and Scholes 380
8.8 Absence of arbitrage and existence of a martingale 381
8.9 How to take a penalty kick, and the minimax theorem 382
8.10 The best lunch and the second welfare theorem 386

Chapter 9. Mathematical Applications 391
9.1 Fun and the quest for the essence 391
9.2 Optimization approach to matrices 392
9.3 How to prove results on linear inequalities 395
9.4 The problem of Apollonius 397
9.5 Minimization of a quadratic function: Sylvester's criterion and Gram's formula 409
9.6 Polynomials of least deviation 411
9.7 Bernstein inequality 414

Chapter 10. Mixed Smooth-Convex Problems 417
10.1 Introduction 417
10.2 Constraints given by inclusion in a cone 419
10.3 Main result: necessary conditions for mixed smooth-convex problems 422
10.4 Proof of the necessary conditions 430
10.5 Discussion and comments 432

Chapter 11. Dynamic Programming in Discrete Time 441
11.0 Summary 441
11.1 Introduction 443
11.2 Main result: Hamilton-Jacobi-Bellman equation 444
11.3 Applications to concrete problems 446
11.4 Exercises 471

Chapter 12. Dynamic Optimization in Continuous Time 475
12.1 Introduction 475
12.2 Main results: necessary conditions of Euler, Lagrange, Pontrya-gin, and Bellman 478
12.3 Applications to concrete problems 492
12.4 Discussion and comments 498

Appendix A. On Linear Algebra: Vector and Matrix Calculus 503
A.1 Introduction 503
A.2 Zero-sweeping or Gaussian elimination, and a formula for the dimension of the solution set 503
A.3 Cramer's rule 507
A.4 Solution using the inverse matrix 508
A.5 Symmetric matrices 510
A.6 Matrices of maximal rank 512
A.7 Vector notation 512
A.8 Coordinate free approach to vectors and matrices 513

Appendix B. On Real Analysis 519
B.1 Completeness of the real numbers 519
B.2 Calculus of differentiation 523
B.3 Convexity 528
B.4 Differentiation and integration 535

Appendix C. The Weierstrass Theorem on Existence of Global Solutions 537
C.1 On the use of the Weierstrass theorem 537
C.2 Derivation of the Weierstrass theorem 544

Appendix D. Crash Course on Problem Solving 547
D.1 One variable without constraints 547
D.2 Several variables without constraints 548
D.3 Several variables under equality constraints 549
D.4 Inequality constraints and convexity 550

Appendix E. Crash Course on Optimization Theory: Geometrical Style 553
E.1 The main points 553
E.2 Unconstrained problems 554
E.3 Convex problems 554
E.4 Equality constraints 555
E.5 Inequality constraints 556
E.6 Transition to infinitely many variables 557

Appendix F. Crash Course on Optimization Theory: Analytical Style 561
F.1 Problem types 561
F.2 Definitions of differentiability 563
F.3 Main theorems of differential and convex calculus 565
F.4 Conditions that are necessary and/or sufficient 567
F.5 Proofs 571

Appendix G. Conditions of Extremum from Fermat to Pontryagin 583
G.1 Necessary first order conditions from Fermat to Pontryagin 583
G.2 Conditions of extremum of the second order 593

Appendix H. Solutions of Exercises of Chapters 1-4 601

Bibliography 645
Index 651

商品描述(中文翻譯)

描述

這本自成一體的教科書是一個非正式的優化入門,通過大量的插圖和應用來介紹優化。重點是通過有限數量的連續變量來解析地解決優化問題。此外,作者還介紹了優化的經典和現代數值方法以及動態優化。

這本書的主要觀點是大多數問題可以通過直接應用費馬、拉格朗日和韋伊斯特拉斯定理來解決。作者展示了如何通過簡單的幾何圖形來支持每個理論結果的直觀理解。他們通過使用各種經典和實際問題的應用來加以說明。即使專家也可能會對其中一些應用感到驚訝。

這本書所涵蓋的主題只需要基本的數學知識即可理解。更高級的讀者,甚至專家,將會驚訝地看到所有主要結果都可以基於費馬-拉格朗日定理來解釋。這本書可以用於連續優化的課程,從入門到高級,適用於任何需要優化的領域。

目錄

前言
0.1 優化:洞察力和應用
0.2 午餐、晚餐和甜點
0.3 這本書是為誰而寫的?
0.4 這本書包含了什麼?
0.5 特點

必要條件:重點是什麼?
第1章 費馬:無約束的單變量
1.0 概述
1.1 引言
1.2 單變量的導數
1.3 主要結果:單變量的費馬定理
1.4 具體問題的應用
1.5 討論和評論
1.6 練習

第2章 費馬:無約束的兩個或多個變量
2.0 概述
2.1 引言
2.2 兩個或多個變量的導數
2.3 主要結果:兩個或多個變量的費馬定理
2.4 具體問題的應用
2.5 討論和評論
2.6 練習

第3章 拉格朗日:等式約束
3.0 概述
3.1 引言
3.2 主要結果:拉格朗日乘數法則
3.3 具體問題的應用
3.4 拉格朗日乘數法則的證明
3.5 討論和評論
3.6 練習

第4章 不等式約束和凸性
4.0 概述
4.1 引言
4.2 主要結果:Karush-Kuhn-Tucker定理
4.3 具體問題的應用
4.4 Karush-Kuhn-Tucker定理的證明
4.5 討論和評論
4.6 練習

第5章 二階條件
5.0 概述
5.1 引言
5.2 主要結果:二階條件
5.3 具體問題的應用
5.4 討論和評論
5.5 練習

第6章 基本算法
6.0 概述
6.1 引言
6.2 非線性優化很困難
6.3 線性優化的主要方法
6.4 線搜索
6.5 下降方向
6.6 近似的質量