## 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. |

### From inside the book

Results 6-10 of 35

In the absence of

**convexity**, we need second order information to tell us more

about minimizers. ... Theorem 2.1.6 (Basic separation) Suppose that the set

**C C E**is closed and convex, and that the point y does not lie in C. Then there

exist a real ...

The

**function**f + e|| || has bounded level

**sets**, so has a global minimizer a by the

Weierstrass proposition (1.1.3). ... (Examples of normal cones) For the following

**sets C C E**, check C is

**convex**and compute the normal cone No(£) for points à in

...

(Normals to affine sets) Given a linear map A : E → Y (where Y is a Euclidean

space) and a point b in Y, prove the normal cone to the ... Hence prove that for a

**convex function**f : C → R and points à, a €

**C C E**, the quotient (f(t+t(a, -3)) – f(x))/t

is ...

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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 |