## 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 1-5 of 43

We say the point x in E is the limit of the

**sequence**of points x', c”, ... in E, written a'

— a as j — co (or limj-ee a = x), if ||a" – c – 0. The closure of D is the set of limits of

**sequences**of points in D, written cl D, and the boundary of D is clD \int D, ...

Theorem 1.1.2 (Bolzano–Weierstrass) Bounded

**sequences**in E have convergent

subsequences. ... Given a set D in E, we call a function f : D → R. continuous (on

D) if f(x') — f(x) for any

**sequence**c' – a in D. In this case it easy to check, ...

Prove 0+(nC,) = n0+(Cy). (d) For a unit vector u in E, prove u e 0+(C) if and only if

there is a

**sequence**(x") in C satisfying ||x"| – Co and |x"|"a" – u. Deduce C is

unbounded if and only if 0" (C) is nontrivial. (e) If Y is a Euclidean space, the map

A ...

Deduce the existence of a

**sequence**(a.”) in C with f(x”) < |x"|/m – +oo. For a fixed

point à in C, derive a contradiction by considering the

**sequence**7m, à -- T.T(x" –

3). |am|| Hence complete the proof of Proposition 1.1.5. The relative interior ...

You have reached your viewing limit for this book.

### What people are saying - Write a review

### 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 |