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More About This Textbook
Overview
This selfcontained 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 FermatLagrange theorem. The book can be used for courses on continuous optimization, from introductory to advanced, for any field for which optimization is relevant.
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.— Jörg Thierfelder
Zentralblatt MATH
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.
— Jörg Thierfelder
Zentralblatt MATH  Jorg Thierfelder
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.Zentralblatt MATH  Jörg Thierfelder
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.From the Publisher
"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."—Jörg Thierfelder, Zentralblatt MATHZentralblatt 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
Product Details
Related Subjects
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.
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: KarushKuhnTucker theorem 204
4.3 Applications to concrete problems 217
4.4 Proof of the KarushKuhnTucker 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 Selfconcordant 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 Arbitragefree 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 SmoothConvex Problems 417
10.1 Introduction 417
10.2 Constraints given by inclusion in a cone 419
10.3 Main result: necessary conditions for mixed smoothconvex 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: HamiltonJacobiBellman 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, Pontryagin, 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 Zerosweeping 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 14 601
Bibliography 645
Index 651