## 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 6-10 of 87

The nonnegative orthant R' is a

**cone**in R” which plays a central role in our

development. In a variety of contexts the analogous role in S” is played by the

**cone**of positive semidefinite matrices, St. (We call a matrix X in S" positive

semidefinite if ...

Prove S' is a closed

**convex cone**with interior S#1. Explain why s: is not a

polyhedron. (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 ...

A

**convex cone**which arises frequently in optimization is the normal cone to a

convex set C at a point à e C, written No.(T). This is the

**convex cone**of normal

vectors, vectors d in E such that (d, a -ā) < 0 for all points a' in C. Proposition 2.1.1

...

In that case local minimizers á may not lie in the interior of the set C of interest, so

the normal

**cone**NC(#) is not simply {0}. The next result shows that when f is

**convex**the first order condition above is sufficient for à to be a global minimizer of

f ...

(Examples of normal

**cones**) For the following sets C C E, check C is

**convex**and

compute the normal

**cone**No(£) for points à in C: (a) C a closed interval in R. (b)

C = B, the unit ball. (c) C a subspace. (d) C a closed halfspace: {a, (a,a) < b) ...

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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 |