## Asymptotic Cones and Functions in Optimization and Variational InequalitiesNonlinear applied analysis and in particular the related ?elds of continuous optimization and variational inequality problems have gone through major developments over the last three decades and have reached maturity. A pivotal role in these developments has been played by convex analysis, a rich area covering a broad range of problems in mathematical sciences and its applications. Separation of convex sets and the Legendre–Fenchel conjugate transforms are fundamental notions that have laid the ground for these fruitful developments. Two other fundamental notions that have contributed to making convex analysis a powerful analytical tool and that haveoftenbeenhiddeninthesedevelopmentsarethenotionsofasymptotic sets and functions. The purpose of this book is to provide a systematic and comprehensive account of asymptotic sets and functions, from which a broad and u- ful theory emerges in the areas of optimization and variational inequa- ties. There is a variety of motivations that led mathematicians to study questions revolving around attaintment of the in?mum in a minimization problem and its stability, duality and minmax theorems, convexi?cation of sets and functions, and maximal monotone maps. In all these topics we are faced with the central problem of handling unbounded situations. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Preface | 1 |

Asymptotic Cones and Functions | 25 |

Existence and Stability in Optimization Problems | 81 |

Minimizing and Stationary Sequences | 119 |

Duality in Optimization Problems | 145 |

Maximal Monotone Maps and Variational Inequalities | 183 |

References | 233 |

243 | |

### Other editions - View all

Asymptotic Cones and Functions in Optimization and Variational Inequalities Alfred Auslender,Marc Teboulle No preview available - 2012 |

Asymptotic Cones and Functions in Optimization and Variational Inequalities Alfred Auslender,Marc Teboulle No preview available - 2013 |

### Common terms and phrases

assumption asymptotic cone asymptotic function asymptotically linear Auslender closed convex set closed set coercive function compact set conjugate conv converging convex analysis convex cone convex optimization convex program Corollary deﬁned deﬁnition denoted domf domh domT dual problem equivalent error bounds exists Fenchel ﬁnite ﬁrst follows fOO(d formula function f Furthermore G dom f G ridom G Rm G Rn given h is lsc holds hypothesis I R U 00 implies inff Lagrangian Lemma Let C C R Let f lev(f linear map lower semicontinuity lsc convex function marginal function maximal monotone maps nonempty and compact obtain optimal set optimal solutions optimization problems primal Proof proper convex function proper function Proposition prove relative interior satisﬁed satisfying semicontinuity sequence xk set-valued map solution set star-monotone subdifferential subdifferential map suﬁiciently large suppose tion variational inequality vector weakly coercive