## 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 83

A set C in E is convex if the line segment joining any two points a and y in C is

contained in C: algebraically, Ax + (1 - A)ye C whenever 0 < As 1. An easy

exercise shows that intersections of

**convex sets**are convex. Given any set D C E,

the ...

In this language the above result states that the point y is separated from the

**set**

C by a hyperplane. In other words, C is contained in a certain closed halfspace

whereas y is not. Thus there is a “dual” representation of C as the intersection of

all ...

Just as for sets, convexity of functions will be crucial for us. Given a

**convex set**

C C E, we say that the function f : C → R is convex if f(\a + (1 – A)y) < \f(x) + (1 - A)f

(y) for all points a, and y in C and 0 < X s. 1. The function f is strictly convex if the ...

Proposition 1.1.5 For a

**convex set**C C E, a convex function f : C – R. has

bounded level sets if and only if it satisfies the growth condition (1.1.4). The proof

is outlined in Exercise 10. Exercises and Commentary Good general references

are ...

(a) Prove the set D – C is closed and convex. (b) Deduce that if in addition D and

C are disjoint then there exists a nonzero element a in E with infreD(a,a) >

supyeo(a,y). Interpret geometrically. (c) Show part (b) fails for the closed

**convex**

**sets**in ...

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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 |