| Preface | p. xi |
| Optimization: insights and applications | p. xiii |
| Lunch, dinner, and dessert | p. xiv |
| For whom is this book meant? | p. xvi |
| What is in this book? | p. xviii |
| Special features | p. xix |
| Necessary Conditions: What Is the Point? | p. 1 |
| Fermat: One Variable without Constraints | p. 3 |
| Summary | p. 3 |
| Introduction | p. 5 |
| The derivative for one variable | p. 6 |
| Main result: Fermat theorem for one variable | p. 14 |
| Applications to concrete problems | p. 30 |
| Discussion and comments | p. 43 |
| Exercises | p. 59 |
| Fermat: Two or More Variables without Constraints | p. 85 |
| Summary | p. 85 |
| Introduction | p. 87 |
| The derivative for two or more variables | p. 87 |
| Main result: Fermat theorem for two or more variables | p. 96 |
| Applications to concrete problems | p. 101 |
| Discussion and comments | p. 127 |
| Exercises | p. 128 |
| Lagrange: Equality Constraints | p. 135 |
| Summary | p. 135 |
| Introduction | p. 138 |
| Main result: Lagrange multiplier rule | p. 140 |
| Applications to concrete problems | p. 152 |
| Proof of the Lagrange multiplier rule | p. 167 |
| Discussion and comments | p. 181 |
| Exercises | p. 190 |
| Inequality Constraints and Convexity | p. 199 |
| Summary | p. 199 |
| Introduction | p. 202 |
| Main result: Karush-Kuhn-Tucker theorem | p. 204 |
| Applications to concrete problems | p. 217 |
| Proof of the Karush-Kuhn-Tucker theorem | p. 229 |
| Discussion and comments | p. 235 |
| Exercises | p. 250 |
| Second Order Conditions | p. 261 |
| Summary | p. 261 |
| Introduction | p. 262 |
| Main result: second order conditions | p. 262 |
| Applications to concrete problems | p. 267 |
| Discussion and comments | p. 271 |
| Exercises | p. 272 |
| Basic Algorithms | p. 273 |
| Summary | p. 273 |
| Introduction | p. 275 |
| Nonlinear optimization is difficult | p. 278 |
| Main methods of linear optimization | p. 283 |
| Line search | p. 286 |
| Direction of descent | p. 299 |
| Quality of approximation | p. 301 |
| Center of gravity method | p. 304 |
| Ellipsoid method | p. 307 |
| Interior point methods | p. 316 |
| Advanced Algorithms | p. 325 |
| Introduction | p. 325 |
| Conjugate gradient method | p. 325 |
| Self-concordant barrier methods | p. 335 |
| Economic Applications | p. 363 |
| Why you should not sell your house to the highest bidder | p. 363 |
| Optimal speed of ships and the cube law | p. 366 |
| Optimal discounts on airline tickets with a Saturday stayover | p. 368 |
| Prediction of flows of cargo | p. 370 |
| Nash bargaining | p. 373 |
| Arbitrage-free bounds for prices | p. 378 |
| Fair price for options: formula of Black and Scholes | p. 380 |
| Absence of arbitrage and existence of a martingale | p. 381 |
| How to take a penalty kick, and the minimax theorem | p. 382 |
| The best lunch and the second welfare theorem | p. 386 |
| Mathematical Applications | p. 391 |
| Fun and the quest for the essence | p. 391 |
| Optimization approach to matrices | p. 392 |
| How to prove results on linear inequalities | p. 395 |
| The problem of Apollonius | p. 397 |
| Minimization of a quadratic function: Sylvester's criterion and Gram's formula | p. 409 |
| Polynomials of least deviation | p. 411 |
| Bernstein inequality | p. 414 |
| Mixed Smooth-Convex Problems | p. 417 |
| Introduction | p. 417 |
| Constraints given by inclusion in a cone | p. 419 |
| Main result: necessary conditions for mixed smooth-convex problems | p. 422 |
| Proof of the necessary conditions | p. 430 |
| Discussion and comments | p. 432 |
| Dynamic Programming in Discrete Time | p. 441 |
| Summary | p. 441 |
| Introduction | p. 443 |
| Main result: Hamilton-Jacobi-Bellman equation | p. 444 |
| Applications to concrete problems | p. 446 |
| Exercises | p. 471 |
| Dynamic Optimization in Continuous Time | p. 475 |
| Introduction | p. 475 |
| Main results: necessary conditions of Euler, Lagrange, Pontryagin, and Bellman | p. 478 |
| Applications to concrete problems | p. 492 |
| Discussion and comments | p. 498 |
| On Linear Algebra: Vector and Matrix Calculus | p. 503 |
| Introduction | p. 503 |
| Zero-sweeping or Gaussian elimination, and a formula for the dimension of the solution set | p. 503 |
| Cramer's rule | p. 507 |
| Solution using the inverse matrix | p. 508 |
| Symmetric matrices | p. 510 |
| Matrices of maximal rank | p. 512 |
| Vector notation | p. 512 |
| Coordinate free approach to vectors and matrices | p. 513 |
| On Real Analysis | p. 519 |
| Completeness of the real numbers | p. 519 |
| Calculus of differentiation | p. 523 |
| Convexity | p. 528 |
| Differentiation and integration | p. 535 |
| The Weierstrass Theorem on Existence of Global Solutions | p. 537 |
| On the use of the Weierstrass theorem | p. 537 |
| Derivation of the Weierstrass theorem | p. 544 |
| Crash Course on Problem Solving | p. 547 |
| One variable without constraints | p. 547 |
| Several variables without constraints | p. 548 |
| Several variables under equality constraints | p. 549 |
| Inequality constraints and convexity | p. 550 |
| Crash Course on Optimization Theory: Geometrical Style | p. 553 |
| The main points | p. 553 |
| Unconstrained problems | p. 554 |
| Convex problems | p. 554 |
| Equality constraints | p. 555 |
| Inequality constraints | p. 556 |
| Transition to infinitely many variables | p. 557 |
| Crash Course on Optimization Theory: Analytical Style | p. 561 |
| Problem types | p. 561 |
| Definitions of differentiability | p. 563 |
| Main theorems of differential and convex calculus | p. 565 |
| Conditions that are necessary and/or sufficient | p. 567 |
| Proofs | p. 571 |
| Conditions of Extremum from Fermat to Pontryagin | p. 583 |
| Necessary first order conditions from Fermat to Pontryagin | p. 583 |
| Conditions of extremum of the second order | p. 593 |
| Solutions of Exercises of Chapters 1-4 | p. 601 |
| Bibliography | p. 645 |
| Index | p. 651 |
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