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Mathematical Programming

, Volume 148, Issue 1–2, pp 241–277 | Cite as

Convex proximal bundle methods in depth: a unified analysis for inexact oracles

  • W. de Oliveira
  • C. SagastizábalEmail author
  • C. Lemaréchal
Full Length Paper Series B

Abstract

The last few years have seen the advent of a new generation of bundle methods, capable to handle inexact oracles, polluted by “noise”. Proving convergence of a bundle method is never simple and coping with inexact oracles substantially increases the technicalities. Besides, several variants exist to deal with noise, each one needing an ad hoc proof to show convergence. We state a synthetic convergence theory, in which we highlight the main arguments and specify which assumption is used to establish each intermediate result. The framework is comprehensive and generalizes in various ways a number of algorithms proposed in the literature. Based on the ingredients of our synthetic theory, we consider various bundle methods adapted to oracles for which high accuracy is possible, yet it is preferable not to make exact calculations often, because they are too time consuming.

Mathematics Subject Classification

90C 49M 65K 

Notes

Acknowledgments

Research partly done during a postdoctoral visit of the first author at Inria. The authors are grateful to the reviewers for many insightful comments.

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  • W. de Oliveira
    • 1
  • C. Sagastizábal
    • 2
    Email author
  • C. Lemaréchal
    • 3
  1. 1.Instituto Nacional de Matemática Pura e Aplicada—IMPARio de JaneiroBrazil
  2. 2.IMPARio de JaneiroBrazil
  3. 3.InriaSaint IsmierFrance

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