## 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 84

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 segment joining any two points a and y in C is contained in C: algebraically, ...

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 ...

Proposition 1.1.3 (Weierstrass) Suppose that the set D C E is nonempty and closed, and that all the level sets of the ... 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) ...

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 ...

(Strong separation) Suppose that the set C C E is closed and convex, and that the set D C E is compact and convex. ... (c) Show part (b) fails for the closed

**convex sets**in R*, D = {x ti > 0, x1a:2 > 1}, C {x | x2 = 0}. 6.

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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 |