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Combinatorial Convexity in Hadamard Manifolds: Existence for Equilibrium Problems

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Abstract

In this paper is introduced a proposal of resolvent for equilibrium problems in terms of the Busemann’s function. A advantage of this new proposal is that, in addition to be a natural extension of its counterpart in the linear setting introduced by Combettes and Hirstoaga (J Nonlinear Convex Anal 6(1): 117–136, 2005), the new term that performs regularization is a convex function in general Hadamard manifolds, being a first step to fully answer to the problem posed by Cruz Neto et al. (J Convex Anal 24(2): 679–684, 2017 Section 5). During our study, some elements of convex analysis are explored in the context of Hadamard manifolds, which are interesting on their own. In particular, we introduce a new definition of convex combination (now commutative) of any finite collection of points and present an associated Jensen-type inequality.

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Acknowledgements

The authors were supported in part by FAPEPI/CNPq, CNPq grants 308330/2018-8 and 406566/2021-6, FAPEG/PRONEM- 201710267000532.

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Authors and Affiliations

  1. Universidade Federal de Goiás, Goias, Brazil

    Glaydston de Carvalho Bento

  2. Universidade Federal do Piauí, Teresina, Piauí, Brazil

    João Xavier Cruz Neto & Ítalo Dowell Lira Melo

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Bento, G.d.C., Cruz Neto, J.X. & Melo, Í.D.L. Combinatorial Convexity in Hadamard Manifolds: Existence for Equilibrium Problems. J Optim Theory Appl 195, 1087–1105 (2022). https://doi.org/10.1007/s10957-022-02112-0

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