## 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 46

The set D is

**bounded**if there is a real k satisfying kB D D, and it is compact if it is

closed and

**bounded**. The following result is a central tool in real analysis.

Theorem 1.1.2 (Bolzano–Weierstrass)

**Bounded**sequences in E have convergent

...

Proposition 1.1.3 (Weierstrass) Suppose that the set D C E is nonempty and

closed, and that all the level sets of the continuous function f : D → R are

**bounded**. Then f has a global minimizer. Just as for sets, convexity of functions

will be crucial ...

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 ...

(Convex growth conditions) (a) Find a function with

**bounded**level sets which

does not satisfy the growth condition (1.1.4). (b) Prove that any function satisfying

(1.1.4) has

**bounded**level sets. (c) Suppose the convex function f : C → R has ...

This result is a precursor of a principle due to Ekeland, which we develop in

Section 7.1. Proposition 2.1.7 If the function f : E → R is differentiable and

**bounded**below then there are points where f has small derivative. Proof. Fix any

real e > 0.

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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 |