## 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 72

The following result is a central tool in

**real**analysis. Theorem 1.1.2 (Bolzano–

Weierstrass) Bounded sequences in E have convergent subsequences. Just as

for sets, geometric and topological ideas also intermingle for the

**functions**we

study.

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, ...

... 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 [156] for elementary

**real**analysis and

[1] ...

(Convex growth conditions) (a) Find a

**function**with bounded level sets which

does not satisfy the growth condition (1.1.4). (b) Prove that ... (Accessibility lemma

) Suppose C is a convex set in E. (a) Prove clo C C + eB for any

**real**e > 0. (b) For

...

Any matrix X in S” has n

**real**eigenvalues (counted by multiplicity), which we write

in nomincreasing order A1(X) > X2(X) > . . . = An(X). In this way we define a

**function**A : S" – R”. We also define a linear map Diag : R” – S", where for a vector

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 |