Optimization: Insights and Applications

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Princeton University Press, Sep 18, 2005 - Business & Economics - 658 pages
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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 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.

 

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Contents

Fermat One Variable without Constraints
3
11 Introduction
5
12 The derivative for one variable
6
Fermat theorem for one variable
14
14 Applications to concrete problems
30
15 Discussion and comments
43
16 Exercises
59
Fermat Two or More Variables without Constraints
85
Mathematical Applications
391
92 Optimization approach to matrices
392
93 How to prove results on linear inequalities
395
94 The problem of Apollonius
397
Sylvesters criterion and Grams formula
409
96 Polynomials of least deviation
411
97 Bernstein inequality
414
Mixed SmoothConvex Problems
417

21 Introduction
87
Fermat theorem for two or more variables
96
24 Applications to concrete problems
101
25 Discussion and comments
127
26 Exercises
128
Lagrange Equality Constraints
135
31 Introduction
138
Lagrange multiplier rule
140
33 Applications to concrete problems
152
34 Proof of the Lagrange multiplier rule
167
35 Discussion and comments
181
36 Exercises
190
Inequality Constraints and Convexity
199
41 Introduction
202
KarushKuhnTucker theorem
204
43 Applications to concrete problems
217
44 Proof of the KarushKuhnTucker theorem
229
45 Discussion and comments
235
46 Exercises
250
Second Order Conditions
261
51 Introduction
262
53 Applications to concrete problems
267
54 Discussion and comments
271
55 Exercises
272
Basic Algorithms
273
61 Introduction
275
62 Nonlinear optimization is difficult
278
63 Main methods of linear optimization
283
64 Line search
286
65 Direction of descent
299
66 Quality of approximation
301
67 Center of gravity method
304
68 Ellipsoid method
307
69 Interior point methods
316
Advanced Algorithms
325
73 Selfconcordant barrier methods
335
Economic Applications
363
82 Optimal speed of ships and the cube law
366
83 Optimal discounts on airline tickets with a Saturday stayover
368
84 Prediction of flows of cargo
370
85 Nash bargaining
373
86 Arbitragefree bounds for prices
378
formula of Black and Scholes
380
88 Absence of arbitrage and existence of a martingale
381
89 How to take a penalty kick and the minimax theorem
382
810 The best lunch and the second welfare theorem
386
102 Constraints given by inclusion in a cone
419
necessary conditions for mixed smoothconvex problems
422
104 Proof of the necessary conditions
430
105 Discussion and comments
432
Dynamic Programming in Discrete Time
441
111 Introduction
443
HamiltonJacobiBellman equation
444
113 Applications to concrete problems
446
114 Exercises
471
Dynamic Optimization in Continuous Time
475
necessary conditions of Euler Lagrange Pontryagin and Bellman
478
123 Applications to concrete problems
492
124 Discussion and comments
498
On Linear Algebra Vector and Matrix Calculus
503
A3 Cramers rule
507
A4 Solution using the inverse matrix
508
A5 Symmetric matrices
510
A6 Matrices of maximal rank
512
A8 Coordinate free approach to vectors and matrices
513
On Real Analysis
519
B2 Calculus of differentiation
523
B3 Convexity
528
B4 Differentiation and Integration
535
The Weierstrass Theorem on Existence of Global Solutions
537
C2 Derivation of the Weierstrass theorem
544
Crash Course on Problem Solving
547
D2 Several variables under equality constraints
548
D3 Several variables under equality constraints
549
D4 Inequality constraints and convexity
550
Crash Course on Optimization Theory Geometrical Style
553
E2 Unconstrained problems
554
E4 Equality constraints
555
E5 Inequality constraints
556
E6 Transition to infinitely many variables
557
Crash Course on Optimization Theory Analytical Style
561
F2 Definitions of differentiability
563
F3 Main theorems of differential and convex calculus
565
F4 Conditions that are necessary andor sufficient
567
F5 Proofs
571
Conditions of Extremum from Fermat to Pontryagin
583
G2 Conditions of extremum of the second order
593
Solutions of Exercises of Chapters 14
601
Bibliography
645
Index
651
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About the author (2005)

Jan Brinkhuis is Associate Professor of Finance and Mathematical Methods and Techniques at the Econometric Institute of Erasmus University. Rotterdam.

Tikhomirov is Deputy Director of the Contemporary Europe Research Centre at The University of Melbourne, Australia.

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