## Foundations of OptimizationThis book covers the fundamental principles of optimization in finite dimensions. It develops the necessary material in multivariable calculus both with coordinates and coordinate-free, so recent developments such as semidefinite programming can be dealt with. |

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### Contents

1 | |

2 Unconstrained Optimization | 31 |

3 Variational Principles | 61 |

4 Convex Analysis | 84 |

5 Structure of Convex Sets and Functions | 117 |

6 Separation of Convex Sets | 140 |

7 Convex Polyhedra | 175 |

8 Linear Programming | 194 |

11 Duality Theory and Convex Programming | 274 |

12 Semiinfinite Programming | 313 |

13 Topics in Convexity | 335 |

14 Three Basic Optimization Algorithms | 361 |

A Finite Systems of Linear Inequalities in VectorSpaces | 407 |

B Descartess Rule of Sign | 413 |

C Classical Proofs of the Open Mapping and Gravess Theorems | 416 |

421 | |

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### Common terms and phrases

ac(C affine set affine space ai(C algebraic algorithms assume Banach spaces compact conjugate-gradient method Consider constraint contradiction convergence convex cone convex function convex programming convex set Corollary critical point Define duality eigenvalues ellipsoid equality follows equation equivalent exists a point Farkas's lemma feasible point finite finite-dimensional Fréchet differentiable function f Gâteaux differentiable gives global minimizer gradient half-spaces Hint hyperplane H implies inequality follows int(C KKT conditions KKT points Lagrangian function Lemma Let f linear functional linear program linear subspace Lipschitz continuous lower semicontinuous matrix maximizer minimization problem minimizer of f multipliers Newton’s method nonlinear programming nonnegative obtain open set optimal solution optimality conditions optimization problem orthogonal polynomial positive definite positive semidefinite Proof quadratic rai(C result ri(C saddle point second-order semi-infinite programming separation theorems sequence Show subset symmetric theory topological vector space zero