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Combinatorial Convexity in Hadamard Manifolds: Existence for Equilibrium Problems

Author

Listed:
  • Glaydston de Carvalho Bento

    (Universidade Federal de Goiás)

  • João Xavier Cruz Neto

    (Universidade Federal do Piauí)

  • Ítalo Dowell Lira Melo

    (Universidade Federal do Piauí)

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.

Suggested Citation

  • Glaydston de Carvalho Bento & João Xavier Cruz Neto & Ítalo Dowell Lira Melo, 2022. "Combinatorial Convexity in Hadamard Manifolds: Existence for Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 1087-1105, December.
    • Handle: RePEc:spr:joptap:v:195:y:2022:i:3:d:10.1007_s10957-022-02112-0
    DOI: 10.1007/s10957-022-02112-0
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    References listed on IDEAS

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    1. Li-wen Zhou & Nan-jing Huang, 2019. "A Revision on Geodesic Pseudo-Convex Combination and Knaster–Kuratowski–Mazurkiewicz Theorem on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 182(3), pages 1186-1198, September.
    2. E. E. A. Batista & G. C. Bento & O. P. Ferreira, 2015. "An Existence Result for the Generalized Vector Equilibrium Problem on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 550-557, November.
    3. Xiangmei Wang & Chong Li & Jen-Chih Yao, 2016. "On Some Basic Results Related to Affine Functions on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 783-803, September.
    4. O. Ferreira & L. Pérez & S. Németh, 2005. "Singularities of Monotone Vector Fields and an Extragradient-type Algorithm," Journal of Global Optimization, Springer, vol. 31(1), pages 133-151, January.
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