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Kamien Schwartz - Dynamic Optimization.pdf
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如题，一本是Kamien Schwartz的动态最优化，主要内容是以variations calculus和optimal control这两种方法，后面也有两章介绍dynamic programming的
另一本更为基础一些，前面内容是静态的最优化的理论，后面两章介绍了动态最优的一些方法和应用。
目录如下：
第一本KS的动态最优：
PART I. CALCULUS OF VARIATIONS
Section 1. Introduction 3
Section 2. Example Solved 12
Section 3. Simplest Problem—Euler Equation 14
Section 4. Examples and Interpretations 21
Section 5. Solving the Euler Equation in Special Cases 30
Section 6. Second Order Conditions 41
Section 7. Isoperimetric Problem 47
Section 8. Free End Value 52
Section 9. Free Horizon—Transversality Conditions 57
Section 10. Equality Constrained Endpoint 65
Section 11. Salvage Value 71
Section 12. Inequality Constraint Endpoints and Sensitivity Analysis 77
Section 13. Corners 86
Section 14. Inequality Constraints in (t , x) 90
Section 15. Infinite Horizon Autonomous Problems 95
Section 16. Most Rapid Approach Paths
Section 17. Diagrammatic Analysis
Section 18. Several Functions and Double Integrals
PART II: OPTIMAL CONTROL
Section 1. Introduction 121
Section 2. Simplest Problem—Necessary Conditions 124
Section 3. Sufficiency 133
Section 4. Interpretations 136
Section 5. Several Variables 142
Section 6. Fixed Endpoint Problems 147
Section 7. Various Endpoint Conditions 155
Section 8. Discounting, Current Values, Comparative Dynamics 164
Section 9. Equilibria in Infinite Horizon Autonomous Problems 174
Section 10. Bounded Controls 185
Section 11. Further Control Constraint 195
Section 12. Discontinuous and Bang-Bang Control 202
Section 13. Singular Solutions and Most Rapid Approach Paths 209
Section 14. The Pontryagin Maximum Principle, Existence 218
Section 15. Further Sufficiency Theorems 221
Section 16. Alternative Formulations 227
Section 17. State Variable Inequality Constraints 230
Section 18. Jumps in the State Variable, Switches in State Equations 240
Section 19. Delayed Response 248
Section 20. Optimal Control with Integral State Equations 253
Section 21. Dynamic Programming 259
Section 22. Stochastic Optimal Control 264
Section 23. Differential Games 272
APPENDIX A. CALCULUS AND NONLINEAR
PROGRAMMING
Section 1. Calculus Techniques 291
Section 2. Mean-Value Theorems 294
Section 3. Concave and Convex Functions 298
Section 4. Maxima and Minima 303
Section 5. Equality Constrained Optimization 307
Section 6. Inequality Constrained Optimization 313
Section 7. Line Integrals and Green's Theorem 320
APPENDIX B. DIFFERENTIAL EQUATIONS
Section 1. Introduction 325
Section 2. Linear First Order Differential Equations 328
Section 3. Linear Second Order Differential Equations 332
Section 4. Linear nth Order Differential Equations 339
Section 5. A Pair of Linear Equations 344
Section 6. Existence and Uniqueness of Solutions 350
第二本：
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 flows 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 prob-
lems 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 di-
mension 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
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