Journal of Global Optimization

, Volume 31, Issue 1, pp 133–151 | Cite as

Singularities of Monotone Vector Fields and an Extragradient-type Algorithm

  • O. P. Ferreira
  • L. R. Lucambio. Pérez
  • S. Z. Németh


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.


extragradient algorithm global optimization Hadamard manifold monotone vector field 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bishop, R.L., O’Neill, B. 1969Manifolds of negative curvatureTrans. Amer. Math. Soc.145149CrossRefMathSciNetMATHGoogle Scholar
  2. Burachik R., Sagastiz l C., Svaiter B.F. (1999). Bundle methods for maximal monotone operators, Ill-posed variational problems and regularization techniques (Trier, 1998), In: Lecture Notes in Econom. and Math. Systems, 477: Springer, Berlin, pp. 49-64.Google Scholar
  3. do Carmo, M.P. 1992Riemannian GeometryBirkhauserBostonCrossRefMATHGoogle Scholar
  4. da Cruz Neto, J.X., Lima, L.L., Oliveira, P.R. 1998Geodesic algorithm in riemannian manifoldsBalkan Journal of Geometry and its Applications.389100MathSciNetMATHGoogle Scholar
  5. da Cruz Neto J.X., Ferreira O.P. and Lucambio Pérez L.R. (2002). Contribution to the study of monotone vectorfields, to be pubished in Acta Mathematica Hungarica.Google Scholar
  6. Ferreira, O.P., Oliveira, P.R. 1998Subgradient algorithm on riemannian manifoldsJournal of Optimization Theory and Applications.9793104CrossRefMathSciNetMATHGoogle Scholar
  7. Gabay, D. 1982Minimizing a differentiable function over a differentiable manifoldsJournal of Optimization Theory and Applications.37177219CrossRefMathSciNetMATHGoogle Scholar
  8. Iusem, A.N. 1994An iterative algorithm for the variational inequality problemComputational and Applied Mathematics.13103114MathSciNetMATHGoogle Scholar
  9. Iusem, A.N., Lucambio, Pérez, Luis, R. 2000An extragradient-type algorithm for non-smooth variational inequalitiesOptimization.48309332CrossRefMathSciNetMATHGoogle Scholar
  10. Iusem, A.N., Svaiter, B.F. 1997A variant of Korpelevich’s method for variational inequalities with a new search strategyOptimization.42309321CrossRefMathSciNetMATHGoogle Scholar
  11. Korpelevich, G.M. 1976The extragradient method for finding saddle points and other problemsEkonomika i Matematcheskie Metody12747756MATHGoogle Scholar
  12. Németh, S.Z. 1999Five kinds of monotone vectorfieldsPU.M.A.9417428Google Scholar
  13. Németh, S.Z. 2001Homeomorphisms and monotone vector fieldsPublicationes Mathematicae58707716MathSciNetMATHGoogle Scholar
  14. Németh, S.Z. 1999Geodesic monotone vectorfieldsLobachevskii Journal of Mathematics.51328MathSciNetMATHGoogle Scholar
  15. Németh, S.Z. 1999Monotone vectorfieldsPublicationes Mathematicae.54437449MATHGoogle Scholar
  16. Németh, S.Z. 1999Monotonicity of the complementarity vector field of a nonexpansive mapActa Mathematica Hungarica.84189197CrossRefMathSciNetMATHGoogle Scholar
  17. Németh S.Z. (2003). Variational inequalities on Hadamard manifolds, to be published in Nonlinear Analysis: Theory, Methods & Applications, 52: 1491-1498Google Scholar
  18. Rapcs, T., Thang, T.T. 1996A class of polynomial variable metric algorithms for linearoptimizationMathematical Programming.74319331CrossRefMathSciNetGoogle Scholar
  19. Rapcs’k, T. 1997Smooth Nonlinear Optimization in RnKluwerAcademic PublishersNew YorkCrossRefGoogle Scholar
  20. Sakai, T. 1996Riemannian Geometry, Translations of Mathematical Monographs 149American Mathematical SocietyProvidence R.IGoogle Scholar
  21. Shiga, K. (1984). Hadamard Manifolds, Advanced Studies in Pure Mathematics 3, Geometry of Geodesics and Related Topic, 239-281.Google Scholar
  22. Solodov, M.V., Svaiter, B.F. 1999A new projection method for variational inequality problemsSIAM J. Control Optim.37765776CrossRefMathSciNetMATHGoogle Scholar
  23. Smith S.T. (1994). Optimization techniques on Riemannian Manifolds, Fields Institute Communications 3, American Mathematical Society, Providence R.I., 113-146Google Scholar
  24. Udri¸ste, C. 1977Continuity of convex functions on Riemannian manifoldsBulletine Mathematique de Roumanie.21215218MathSciNetGoogle Scholar
  25. Udri¸ste, C. (1994). Convex Functions and Optimization Methods on Riemannian Manifolds, Mathematics and its Applications 297, KluwerAcademic Publishers.Google Scholar
  26. Udri¸ste, C. 1976Convex functions on Riemannian manifoldsSt. Cerc. Mat.28735745MathSciNetGoogle Scholar
  27. Walter, R. 1974On the metric projections onto convex sets in Riemannian spacesArch. Math.XXV9198CrossRefGoogle Scholar
  28. Zeidler, E. (1990). Nonlinear Functional Analysis and its Applications II/B: Nonlinear Monotone Operators, Springer Verlag.Google Scholar

Copyright information

© Springer Science+Business Media New York 2005

Authors and Affiliations

  • O. P. Ferreira
    • 1
  • L. R. Lucambio. Pérez
    • 1
  • S. Z. Németh
    • 2
  1. 1.IMEUniversidade Federal de GoiásGoâniaBrazil
  2. 2.Computer and Automation InstituteHungarian Academy of SciencesHungary

Personalised recommendations