Optimization: Insights and Applications: Insights and Applications [NOOK Book]

Overview

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 ...

See more details below
Optimization: Insights and Applications: Insights and Applications

Available on NOOK devices and apps  
  • NOOK Devices
  • NOOK HD/HD+ Tablet
  • NOOK
  • NOOK Color
  • NOOK Tablet
  • Tablet/Phone
  • NOOK for Windows 8 Tablet
  • NOOK for iOS
  • NOOK for Android
  • NOOK Kids for iPad
  • PC/Mac
  • NOOK for Windows 8
  • NOOK for PC
  • NOOK for Mac
  • NOOK Study
  • NOOK for Web

Want a NOOK? Explore Now

NOOK Book (eBook - Course Book)
$65.99
BN.com price
(Save 42%)$115.00 List Price

Overview

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.

Read More Show Less

Editorial Reviews

Zentralblatt MATH Database
The authors provide a very nice and interesting textbook on the theory and the application of mathematical optimization. . . . The book is written as well as for beginners and for experts. . . . Both types of readers can profit from the given shortcuts and royal roads which jump over some theoretical explanations and lead directly to the applications.
— Jorg Thierfelder
Read More Show Less

Product Details

  • ISBN-13: 9781400829361
  • Publisher: Princeton University Press
  • Publication date: 2/11/2011
  • Series: Princeton Series in Applied Mathematics
  • Sold by: Barnes & Noble
  • Format: eBook
  • Edition description: Course Book
  • Pages: 688
  • File size: 26 MB
  • Note: This product may take a few minutes to download.

Meet the Author

Jan Brinkhuis is Associate Professor of Finance and Mathematical Methods and Techniques at the Econometric Institute of Erasmus University, Rotterdam. Vladimir Tikhomirov holds the Chair of Optimal Control in the Department of Mechanics and Mathematics at the Lomonosov Moscow State University.
Read More Show Less

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

Read More Show Less

Customer Reviews

Be the first to write a review
( 0 )
Rating Distribution

5 Star

(0)

4 Star

(0)

3 Star

(0)

2 Star

(0)

1 Star

(0)

Your Rating:

Your Name: Create a Pen Name or

Barnes & Noble.com Review Rules

Our reader reviews allow you to share your comments on titles you liked, or didn't, with others. By submitting an online review, you are representing to Barnes & Noble.com that all information contained in your review is original and accurate in all respects, and that the submission of such content by you and the posting of such content by Barnes & Noble.com does not and will not violate the rights of any third party. Please follow the rules below to help ensure that your review can be posted.

Reviews by Our Customers Under the Age of 13

We highly value and respect everyone's opinion concerning the titles we offer. However, we cannot allow persons under the age of 13 to have accounts at BN.com or to post customer reviews. Please see our Terms of Use for more details.

What to exclude from your review:

Please do not write about reviews, commentary, or information posted on the product page. If you see any errors in the information on the product page, please send us an email.

Reviews should not contain any of the following:

  • - HTML tags, profanity, obscenities, vulgarities, or comments that defame anyone
  • - Time-sensitive information such as tour dates, signings, lectures, etc.
  • - Single-word reviews. Other people will read your review to discover why you liked or didn't like the title. Be descriptive.
  • - Comments focusing on the author or that may ruin the ending for others
  • - Phone numbers, addresses, URLs
  • - Pricing and availability information or alternative ordering information
  • - Advertisements or commercial solicitation

Reminder:

  • - By submitting a review, you grant to Barnes & Noble.com and its sublicensees the royalty-free, perpetual, irrevocable right and license to use the review in accordance with the Barnes & Noble.com Terms of Use.
  • - Barnes & Noble.com reserves the right not to post any review -- particularly those that do not follow the terms and conditions of these Rules. Barnes & Noble.com also reserves the right to remove any review at any time without notice.
  • - See Terms of Use for other conditions and disclaimers.
Search for Products You'd Like to Recommend

Recommend other products that relate to your review. Just search for them below and share!

Create a Pen Name

Your Pen Name is your unique identity on BN.com. It will appear on the reviews you write and other website activities. Your Pen Name cannot be edited, changed or deleted once submitted.

 
Your Pen Name can be any combination of alphanumeric characters (plus - and _), and must be at least two characters long.

Continue Anonymously

    If you find inappropriate content, please report it to Barnes & Noble
    Why is this product inappropriate?
    Comments (optional)