## Optimization: Insights and ApplicationsThis 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 |

645 | |

651 | |