## 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 70

However, rather like Halmos's Finite Dimensional

**Vector**Spaces [81], ease of

extension beyond finite dimensions substantially motivates our choice of

approach. Where possible, we have chosen a proof technique permitting those

readers ...

Our setting, for most of the book, is an arbitrary Euclidean space E, by which we

mean a finite-dimensional

**vector**space over ... We would lose no generality if we

considered only the space R” of real (column) n-

**vectors**(with its standard inner ...

and the cone of

**vectors**with nonincreasing components # = {x e R”|al > x2 > . . . 2

an}. The smallest cone containing a given set D C E is clearly R+D. The

fundamental geometric idea of this book is convexity. A set C in E is convex if the

line ...

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 ...

Xx > 0 for all

**vectors**x in R”, and positive definite if the inequality is strict

whenever a is nonzero.) These two cones have some important differences; in

particular, R' is a polyhedron, whereas the cone of positive semidefinite matrices

S' is not, ...

### 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 |