## 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

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 ...

(

**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.

(b)

**Prove**that any function satisfying (1.1.4) has bounded level sets. (c) Suppose the convex function

**f**: C → R has bounded level sets but that (1.1.4) fails. Deduce the existence of a sequence (a.”) in C with

**f**(x”) < |x"|/m – +oo.

(e) For any point a in D,

**prove**affD = x+span (D–a), and deduce the linear subspace span (D - a) is independent of a. ... (e) If

**F**is another Euclidean space and the map A : E –

**F**is linear,

**prove**riAC D Ari C. 1.2 Symmetric Matrices ...

Theorem 2.1.5 (Second order conditions) Suppose the twice continuously differentiable function

**f**: R” – R has a critical point E. If E is a local ... A good illustration is the separation result of Section 1.1, which we now

**prove**.

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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 |