## 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 54

**Hence**our aim of writing a concise, accessible account of convex analysis and its

applications and extensions, for a broad audience. For students of optimization

and analysis, there is great benefit to blurring the distinction between the two ...

To ensure these are always defined, it is natural to append -oo and +oo to the

real numbers, and allow their use in the usual notation for open and closed

intervals.

**Hence**, inf() = +oo and sup() = -oo, and for example (—co, +oo] denotes

the ...

|am||

**Hence**complete the proof of Proposition 1.1.5. The relative interior Some

arguments about finite-dimensional convex sets C simplify and lose no generality

if we assume C contains 0 and spans E. The following exercises outline this idea.

(b) Suppose dim E > 0, 0 e C and aff C = E. Prove C contains a basis {x', c*, ..., x”}

of E. Deduce (1/(n + 1))YU'a e int C.

**Hence**deduce that any nonempty convex set

in E has nonempty relative interior. (c) Prove that for 0 < A s 1 we have AriC + ...

(Singular values and von Neumann's lemma) Let M” denote the vector space of n

×n real matrices. For a matrix A in M” we define the singular values of A by g(A) =

V/X(ATA) for i = 1,2,...,n, and

**hence**define a map a : M” – R”. (Notice zero may ...

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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 |