## 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 82

Just as for sets, geometric and topological ideas also intermingle for the functions

we study. Given a set D in E, we call a

**function f**: D → R. continuous (on D) if f(x')

— f(x) for any sequence c' – a in D. In this case it easy to check, for example, ...

A (global) minimizer of a

**function f**: D → R is a point à in D at which f attains its

infimum inff = inf f(D) = inf{f(x)|r e D}. In this case we refer to £ as an optimal

solution of the optimization problem info f. For a positive real 6 and a function g :(

0, ...

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 ... The theory of the relative interior

(Exercises 11, 12, and 13) is developed extensively in [149 (which is also a good

...

(f) Consider another nonempty closed convex set D C E such that 0+(C) n0+(D) is

a linear subspace. Prove C–D is closed. 7. For any set of vectors a', a”, ..., a” in E,

prove the

**function f**(x) = max: (a', a) is convex on E. 8. Prove Proposition 1.1.3 ...

(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 ...

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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 |