## Introduction to Optimization and Semidifferential CalculusThis text provides a modern and mathematically rigorous treatment of semidifferential calculus in the context of optimization. Semidifferentials are a natural tool for solving certain problems in non-differentiable optimization. Classical notions in convex analysis are introduced (convexification, duality, linear and quadratic programming, two-person zero-sum games, Lagrange primal and dual problems, semiconvex and semiconcave functions) and the theory is developed to a sophisticated enough level to tackle finite-dimensional versions of problems in the calculus of variations. This text is designed for a one-term course at the undergraduate level in a wide variety of numerate disciplines and includes sufficient background material in calculus and linear algebra to be self-contained. Additional material beyond the scope of a basic undergraduate course is included to further develop the theory. The theoretical content of the text is enriched by numerous examples and exercises, for which solutions are included. |

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### Contents

MO12_ch1 | 1 |

MO12_ch2 | 11 |

MO12_ch3 | 67 |

MO12_ch4 | 153 |

MO12_ch5 | 241 |

MO12_appa | 291 |

MO12_appb | 295 |

MO12_bm | 339 |

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argmin f(U bounded calculus calculus of variations Chapter characterization choose closed convex cone compact constraint functions convex cone convex function convex set d f(x defined Dg(x dH f(x;v differential quotient equivalent Example exists f is continuous f is convex f is Fréchet f is lsc Fenchel–Legendre transform finite Fºl Fréchet differentiable function f Gateaux differentiable growth property H f(x Hadamard semidifferentiable hence Hessian matrix inequality inf f(U infimum Lagrange multipliers Lemma Let f lim Sup liminf linear subspace lower semicontinuity minimizer of f necessary optimality condition neighborhood V(x nonempty notion objective function point x0 positive definite Proof prove regular point RU{+oo semicontinuity semiconvex sequence ſº strictly convex subset symmetric matrix Theorem 3.1 TV x0 U C R variables vector