## Convex Analysis and Nonlinear Optimization: Theory and ExamplesOptimization is a rich and thriving mathematical discipline. The theory underlying current computational optimization techniques grows ever more sophisticated. The powerful and elegant language of convex analysis unifies much of this theory. The aim of this book is to provide a concise, accessible account of convex analysis and its applications and extensions, for a broad audience. It can serve as a teaching text, at roughly the level of first year graduate students. While the main body of the text is self-contained, each section concludes with an often extensive set of optional exercises. The new edition adds material on semismooth optimization, as well as several new proofs that will make this book even more self-contained. |

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Results 1-5 of 68

If C is

**nonempty**and satisfies R+C = C we call it a cone. (Notice we require that

cones contain the origin.) Examples are the positive orthant R# = {x e R" | each x

> 0}, and the cone of vectors with nonincreasing components # = Background ...

Proposition 1.1.3 (Weierstrass) Suppose that the set D C E is

**nonempty**and

closed, and that all the level sets of the continuous function f : D → R are

bounded. Then f has a global minimizer. Just as for sets, convexity of functions

will be crucial ...

(Recession cones) Consider a

**nonempty**closed convex set C C E. We define the

recession cone of C by 0+(C) = {d e E|C + R+d c C}. (a) Prove 0" (C) is a closed

convex cone. (b) Prove d e 0+(C) if and only if a + R+d C C for some point a in C.

(e) Deduce further that if int C is

**nonempty**then cl(int C) = cl C. Is convexity

necessary? 12. " (Affine sets) A set L in E is affine if the entire line through any

distinct points a and y in L lies in L: algebraically, Xa: + (1 – X)y € L for any real A.

The ...

(b) Suppose dim E > 0, 0 e C and aff C = E. Prove C contains a basis {x', c*, ..., x”}

of E. Deduce (1/(n + 1))YU'a e int C. Hence deduce that any

**nonempty**convex set

in E has

**nonempty**relative interior. (c) Prove that for 0 < A s 1 we have AriC + ...

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### Contents

15 | |

Fenchel Duality | 33 |

Convex Analysis | 65 |

Special Cases | 97 |

Nonsmooth Optimization | 123 |

KarushKuhnTucker Theory | 153 |

Fixed Points | 179 |

Infinite Versus Finite Dimensions | 209 |

List of Results and Notation | 221 |

Bibliography | 241 |

Index | 253 |

### Other editions - View all

Convex Analysis and Nonlinear Optimization: Theory and Examples Jonathan M. Borwein,Adrian S. Lewis No preview available - 2000 |