Stationarity and Convergence in Reduce-or-Retreat Minimization

  • Adam B. Levy

Part of the SpringerBriefs in Optimization book series (BRIEFSOPTI)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Adam B. Levy
    Pages 13-42
  3. Adam B. Levy
    Pages 43-53
  4. Back Matter
    Pages 55-55

About this book

Introduction

​​​​​​ Stationarity and Convergence in Reduce-or-Retreat Minimization presents and analyzes a unifying framework for a wide variety of numerical methods in optimization. The author’s “reduce-or-retreat” framework is a conceptual method-outline that covers any method whose iterations choose between reducing the objective in some way at a trial point, or retreating to a closer set of trial points. The alignment of various derivative-based methods within the same framework encourages the construction of new methods, and inspires new theoretical developments as companions to results from across traditional divides. The text illustrates the former by developing two generalizations of classic derivative-based methods which accommodate non-smooth objectives, and the latter by analyzing these two methods in detail along with a pattern-search method and the famous Nelder-Mead method.In addition to providing a bridge for theory through the “reduce-or-retreat” framework, this monograph extends and broadens the traditional convergence analyses in several ways. Levy develops a generalized notion of approaching stationarity which applies to non-smooth objectives, and explores the roles of the descent and non-degeneracy conditions in establishing this property. The traditional analysis is broadened by considering “situational” convergence of different elements computed at each iteration of a reduce-or-retreat method. The “reduce-or-retreat” framework described in this text covers specialized minimization methods, some general methods for minimization and a direct search method, while providing convergence analysis which complements and expands existing results.​ ​

Keywords

convergence analysis derivative-free methods non-smooth analysis numerical methods reduce or retreat methods in optimization

Authors and affiliations

  • Adam B. Levy
    • 1
  1. 1.Bowdoin CollegeBrunswickUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4614-4642-2
  • Copyright Information Adam B. Levy 2012
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4614-4641-5
  • Online ISBN 978-1-4614-4642-2
  • Series Print ISSN 2190-8354
  • Series Online ISSN 2191-575X
  • About this book