## 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 87

Hence our aim of writing a concise, accessible account of

**convex**analysis and its

applications and extensions, for a broad ... and semidefinite programming duality

and

**cone**polarity), we constantly emphasize the power of abstract models and ...

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 ...

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 ...

Separation theorems for

**convex**sets originate with Minkowski [129). The theory

of the relative interior (Exercises 11, 12, and 13) is developed extensively in [149

(which is also a good reference for the recession

**cone**, Exercise 6). 1. Prove the ...

(Strong separation) Suppose that the set C C E is closed and

**convex**, and that

the set D C E is compact and

**convex**. ... (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

+ ...

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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 |