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Maximal Monotone Inclusions and Fitzpatrick Functions

Author

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  • J. M. Borwein

    (University of Newcastle)

  • J. Dutta

    (Indian Institute of Technology Kanpur)

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.

Suggested Citation

  • J. M. Borwein & J. Dutta, 2016. "Maximal Monotone Inclusions and Fitzpatrick Functions," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 757-784, December.
    • Handle: RePEc:spr:joptap:v:171:y:2016:i:3:d:10.1007_s10957-015-0813-x
    DOI: 10.1007/s10957-015-0813-x
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    References listed on IDEAS

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    1. D. Aussel & J. Dutta, 2011. "On Gap Functions for Multivalued Stampacchia Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 149(3), pages 513-527, June.
    2. NESTEROV, Yurii & SCRIMALI, Laura, 2011. "Solving strongly monotone variational and quasi-variational inequalities," LIDAM Reprints CORE 2357, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Panayotis Mertikopoulos & Mathias Staudigl, 2018. "Stochastic Mirror Descent Dynamics and Their Convergence in Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 838-867, December.
    2. Shisheng Cui & Uday Shanbhag & Mathias Staudigl & Phan Vuong, 2022. "Stochastic relaxed inertial forward-backward-forward splitting for monotone inclusions in Hilbert spaces," Computational Optimization and Applications, Springer, vol. 83(2), pages 465-524, November.

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