## 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 94

... any point x in C, the directional derivative, if it exists, satisfies f'(x;a – 3) > 0. In

particular, if f is differentiable at E, then the condition —Vf(x) e NC(x) holds.

**Proof**.

If some point a in C satisfies f'(3; a 15 Inequality Constraints Optimality Conditions

.

**Proof**. If some point a in C satisfies f'(3; a – 3) < 0, then all small real t > 0 satisfy f(

x + t(a – 3)) < f(x), but this contradicts the local minimality of ā. D The case of this

result where C is an open set is the canonical introduction to the use of calculus ...

Theorem 2.1.6 (Basic separation) Suppose that the set C C E is closed and

convex, and that the point y does not lie in C. Then there exist a real b and a

nonzero element a of E such that (a, y) > b > (a,a) for all points a in C.

**Proof**. We

may ...

**Proof**. Fix any real e > 0. The function f + e|| || has bounded level sets, so has a

global minimizer a by the Weierstrass proposition (1.1.3). If the vector d = Vf(x*)

satisfies ||d|| > 6 then, from the inequality +--vie),4)--a <-la. we would have for

small t > ...

(e) Find an alternative

**proof**of part (d) by using a spectral decomposition of A. (

Another approach to this problem is given in Section 7.2, Exercise 6.) 7. Suppose

a convex function g : [0, 1] – R satisfies g(0) = 0. Prove the function t e (0, ...

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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 |