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Singularities of Monotone Vector Fields and an Extragradient-type Algorithm

Author

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  • O. Ferreira
  • L. Pérez
  • S. Németh

Abstract

Bearing in mind the notion of monotone vector field on Riemannian manifolds, see [12--16], we study the set of their singularities and for a particularclass of manifolds develop an extragradient-type algorithm convergent to singularities of such vector fields. In particular, our method can be used forsolving nonlinear constrained optimization problems in Euclidean space, with a convex objective function and the constraint set a constant curvature Hadamard manifold. Our paper shows how tools of convex analysis on Riemannian manifolds can be used to solve some nonconvex constrained problem in a Euclidean space. Copyright Springer Science+Business Media New York 2005

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:31:y:2005:i:1:p:133-151
    DOI: 10.1007/s10898-003-3780-y
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    Citations

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    Cited by:

    1. Li-Wen Zhou & Nan-Jing Huang, 2013. "Existence of Solutions for Vector Optimization on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 44-53, April.
    2. Xiao-bo Li & Li-wen Zhou & Nan-jing Huang, 2016. "Gap Functions and Global Error Bounds for Generalized Mixed Variational Inequalities on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 830-849, March.
    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. 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.
    5. Glaydston Carvalho Bento & João Xavier Cruz Neto & Paulo Roberto Oliveira, 2016. "A New Approach to the Proximal Point Method: Convergence on General Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 743-755, March.
    6. Guo-ji Tang & Nan-jing Huang, 2012. "Korpelevich’s method for variational inequality problems on Hadamard manifolds," Journal of Global Optimization, Springer, vol. 54(3), pages 493-509, November.
    7. Konrawut Khammahawong & Parin Chaipunya & Kamonrat Sombut, 2022. "Approximating Common Fixed Points of Nonexpansive Mappings on Hadamard Manifolds with Applications," Mathematics, MDPI, vol. 10(21), pages 1-20, November.
    8. G. C. Bento & J. X. Cruz Neto & P. A. Soares & A. Soubeyran, 2022. "A new regularization of equilibrium problems on Hadamard manifolds: applications to theories of desires," Annals of Operations Research, Springer, vol. 316(2), pages 1301-1318, September.
    9. G. C. Bento & J. X. Cruz Neto, 2013. "A Subgradient Method for Multiobjective Optimization on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 159(1), pages 125-137, October.
    10. João Carlos de O. Souza, 2018. "Proximal Point Methods for Lipschitz Functions on Hadamard Manifolds: Scalar and Vectorial Cases," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 745-760, December.
    11. X. M. Wang & C. Li & J. C. Yao, 2015. "Subgradient Projection Algorithms for Convex Feasibility on Riemannian Manifolds with Lower Bounded Curvatures," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 202-217, January.
    12. J. H. Wang & G. López & V. Martín-Márquez & C. Li, 2010. "Monotone and Accretive Vector Fields on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 691-708, September.
    13. J. H. Wang, 2011. "Convergence of Newton’s Method for Sections on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 148(1), pages 125-145, January.
    14. J. X. Cruz Neto & F. M. O. Jacinto & P. A. Soares & J. C. O. Souza, 2018. "On maximal monotonicity of bifunctions on Hadamard manifolds," Journal of Global Optimization, Springer, vol. 72(3), pages 591-601, November.
    15. Li-wen Zhou & Yi-bin Xiao & Nan-jing Huang, 2017. "New Characterization of Geodesic Convexity on Hadamard Manifolds with Applications," Journal of Optimization Theory and Applications, Springer, vol. 172(3), pages 824-844, March.
    16. X. M. Wang & J. H. Wang & C. Li, 2023. "Convergence of Inexact Steepest Descent Algorithm for Multiobjective Optimizations on Riemannian Manifolds Without Curvature Constraints," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 187-214, July.
    17. Jin-Hua Wang & Jen-Chih Yao & Chong Li, 2012. "Gauss–Newton method for convex composite optimizations on Riemannian manifolds," Journal of Global Optimization, Springer, vol. 53(1), pages 5-28, May.
    18. 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.

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