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Journal of Optimization Theory and Applications

, Volume 171, Issue 3, pp 757–784 | Cite as

Maximal Monotone Inclusions and Fitzpatrick Functions

  • J. M. Borwein
  • J. DuttaEmail author
Article

Abstract

In this paper, we study maximal monotone inclusions from the perspective of gap functions. We propose a very natural gap function for an arbitrary maximal monotone inclusion and will demonstrate how naturally this gap function arises from the Fitzpatrick function, which is a convex function, used to represent maximal monotone operators. This allows us to use the powerful strong Fitzpatrick inequality to analyse solutions of the inclusion. We also study the special cases of a variational inequality and of a generalized variational inequality problem. The associated notion of a scalar gap is also considered in some detail. Corresponding local and global error bounds are also developed for the maximal monotone inclusion.

Keywords

Maximal monotone operator Monotone inclusions Variational inequality Fitzpatrick function Gap functions  Error bounds 

Mathematics Subject Classification

90C30 49J52 

Notes

Acknowledgments

We are thankful to the anonymous referees for their constructive suggestions which has improved the presentation of the paper and also bringing to our notice the references [12] and [17]. We would also like to thank Poonam Kesarwani for her help with the MATLAB.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.CARMAUniversity of NewcastleCallaghanAustralia
  2. 2.Economics Group, Department of Humanities and Social ScienceIndian Institute of Technology KanpurKanpurIndia

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