## 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. |

### From inside the book

Results 1-5 of 47

In fact any linear function from E to R has the form (a, -) for some element a of E.

**Linear maps**and affine functions (linear ... The adjoint of the map A above is the

**linear map**A* : Y → E defined by the property (A"y, a) = (y, Aa) for all points a in E

...

(e) If Y is a Euclidean space, the

**map**A : E → Y is

**linear**, and N(A)n 0+(C) is a

**linear**subspace, prove AC is closed. Show this result can fail without the last

assumption. (f) Consider another nonempty closed convex set D C E such that 0

+(C) ...

(ii) For any point y in C there exists a real e > 0 with a +é(a:-y) in C. (iii) R+(C - a)

is a

**linear**subspace. (e) If F is another Euclidean space and the

**map**A : E – F is

**linear**, prove riAC D Ari C. 1.2 Symmetric Matrices Throughout most of this book ...

We also define a

**linear map**Diag : R” – S", where for a vector a in R”, Diaga is an

n x n diagonal matrix with diagonal entries vi. This map embeds R” as a

subspace of S" and the cone R' as a subcone of S'. The determinant of a square

matrix Z ...

For a fixed column vectors in R”, define a

**linear map**A : S" – R.” by setting AX =

Xs for any matrix X in S". Calculate the adjoint map A”. * (Fan's inequality) For

vectors x and y in R” and a matrix U in O”, define a = (Diaga, U"(Diagy)0). (a)

Prove a ...

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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 |