## 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 36

Just as for sets, convexity of functions will be crucial for us. Given a convex set

C C E, we say that the function f : C → R is convex if f(\a + (1 – A)y) < \f(x) + (1 - A)f

(y) for all points a, and y in C and 0 < X s. 1. The function f is

**strictly convex**if the ...

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). ... (b) We

see later (Theorem 3.1.11) that the function – log is

**convex**on the

**strictly**positive

reals.

(S: is not

**strictly convex**) Find nonzero matrices X and Y in Sł such that RLX #

REY and (X +Y)/2 g S$4. (A nonlattice ordering) Suppose the matrix Z in S

satisfies 0 0 0 1 1 0 w=|| || |and w=| | <= W = Z. (a) By considering diagonal W,

prove 1 a 2- ...

(Nearest points) (a) Prove that if a function f : C → R is

**strictly convex**then it has

at most one global minimizer on C. (b) Prove the function f(x) = |x-y|*/2 is

**strictly**

**convex**on E for any point y in E. (c) Suppose C is a nonempty, closed convex ...

(c) Prove the power function p is

**strictly convex**. (d) Use part (a) of Exercise 8 to

show that the conservation of current equations in part (b) have a unique solution

. ** (Matrix completion [77]) For a set A c{(i,j)|1 < i < j < n}, suppose the subspace

...

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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 |