Skip to main content
SpringerLink
Log in

Monotone and Accretive Vector Fields on Riemannian Manifolds

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The relationship between monotonicity and accretivity on Riemannian manifolds is studied in this paper and both concepts are proved to be equivalent in Hadamard manifolds. As a consequence an iterative method is obtained for approximating singularities of Lipschitz continuous, strongly monotone mappings. We also establish the equivalence between the strong convexity of functions and the strong monotonicity of its subdifferentials on Riemannian manifolds. These results are then applied to solve the minimization of convex functions on Riemannian manifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 73, 875–881 (1967)

    MathSciNet  MATH  Google Scholar 

  2. Brezis, H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert. American Elsevier Publishing, New York (1973)

    MATH  Google Scholar 

  3. Zeidler, E.: Nonlinear Functional Analysis and Applications. II/B. Nonlinear Monotone Operators. Springer, New York (1990)

    MATH  Google Scholar 

  4. Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht (1990)

    MATH  Google Scholar 

  5. Da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R.: Monotone point-to-set vector fields. Balk. J. Geom. Appl. 5, 69–79 (2000)

    MATH  Google Scholar 

  6. Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51, 257–270 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ferreira, O.P., Lucambio Pérez, L.R., Nemeth, S.Z.: Singularities of monotone vector fields and an extragradient-type algorithm. J. Glob. Optim. 31, 133–151 (2005)

    Article  MATH  Google Scholar 

  8. Rapcsák, T.: Smooth Nonlinear Optimization in ℝn. Nonconvex Optimization and Its Applications, vol. 19. Kluwer Academic, Dordrecht (1997)

    Google Scholar 

  9. Martín-Márquez, V.: Nonexpansive mappings and monotone vector fields in Hadamard manifold. Commun. Appl. Anal. 13, 633–646 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Smith, S.T.: Optimization Techniques on Riemannian Manifolds. Fields Institute Communications, vol. 3, pp. 113–146. American Mathematical Society, Providence (1994)

    Google Scholar 

  11. Sturm, K.T.: Probability measures on metric spaces of nonpositive curvature. In: Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Paris, 2002. Contemp. Math., vol. 338, pp. 357–390. Am. Math. Soc., Providence (2003)

    Google Scholar 

  12. Iwamiya, T., Okochi, H.: Monotonicity, resolvents and Yosida approximations of operators on Hilbert manifolds. Nonlinear Anal. 54, 205–214 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and Its Applications, vol. 297. Kluwer Academic, Dordrecht (1994)

    MATH  Google Scholar 

  14. Li, S.L., Li, C., Liou, Y.C., Yao, J.C.: Existence of solutions for variational inequalities on Riemannian manifolds. Nonlinear Anal. 71(11), 5695–5706 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ledyaev, Y.S., Zhu, Q.J.: Nonsmooth analysis on smooth manifolds. Trans. Am. Math. Soc. 359, 3687–3732 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dedieu, J.P., Priouret, P., Malajovich, G.: Newton’s method on Riemannian manifolds: covariant alpha theory. IMA J. Numer. Anal. 23, 395–419 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ferreira, O.P., Svaiter, B.F.: Kantorovich’s theorem on Newton’s method in Riemannian manifolds. J. Complex. 18, 304–329 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, C., Wang, J.H.: Convergence of the Newton method and uniqueness of zeros of vector fields on Riemannian manifolds. Sci. China Ser. A 48, 1465–1478 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, C., Wang, J.H.: Newton’s method on Riemannian manifolds: Smale’s point estimate theory under the γ-condition. IMA J. Numer. Anal. 26, 228–251 (2006)

    Article  MathSciNet  Google Scholar 

  20. Li, C., Wang, J.H.: Newton’s method for sections on Riemannian manifolds: generalized covariant α-theory. J. Complex. 24, 423–451 (2008)

    Article  MATH  Google Scholar 

  21. Wang, J.H., Li, C.: Uniqueness of the singular points of vector fields on Riemannian manifolds under the γ-condition. J. Complex. 22, 533–548 (2006)

    Article  MATH  Google Scholar 

  22. Azagra, D., Ferrera, J., López-Mesas, F.: Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds. J. Funct. Anal. 220, 304–361 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, C., López, G., Martín-Márquez, V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79(3), 663–683 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Németh, S.Z.: Monotone vector fields. Publ. Math. Debr. 54(3–4), 437–449 (1999)

    MATH  Google Scholar 

  25. Németh, S.Z.: Geodesic monotone vector fields. Lobachevskii J. Math. 5, 13–28 (1999)

    MathSciNet  MATH  Google Scholar 

  26. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  27. DoCarmo, M.P.: Riemannian Geometry. Birkhauser, Boston (1992)

    Google Scholar 

  28. Sakai, T.: Riemannian Geometry. Translations of Mathematical Monographs, vol. 149. American Mathematical Society, Providence (1996)

    MATH  Google Scholar 

  29. Walter, R.: On the metric projection onto convex sets in Riemannian spaces. Arch. Math. 25, 91–98 (1974)

    Article  MATH  Google Scholar 

  30. Cheeger, J.D., Gromoll, D.: On the structure of complete manifolds of nonnegative curvature. Ann. Math. 96(2), 413–443 (1972)

    Article  MathSciNet  Google Scholar 

  31. Chidume, C.E.: Iterative approximation of fixed points of Lipschitzian strictly pseudo-contractive mappings. Proc. Am. Math. Soc. 99(2), 283–288 (1987)

    MathSciNet  MATH  Google Scholar 

  32. Chidume, C.E.: Iterative approximation of the solution of a monotone operator equation in certain Banach spaces. International Centre for Theorical Physics IC/88/22 (1988)

  33. Kamimura, S., Takahashi, W.: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 13, 226–240 (2000)

    Article  MathSciNet  Google Scholar 

  34. Polyak, B.T.: Existence theorems and convergence of minimizing sequences in extremun problems with restrictions. Dokl. Akad. Nauk. USSR 166, 287–290 (1966)

    Google Scholar 

  35. Li, C., Mordukhovich, B.S., Wang, J.H., Yao, J.C.: Weak sharp minima on Riemannian manifolds. SIAM J. Optim. (submitted)

  36. Da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R.: Contributions to the study of monotone vector fields. Acta Math. Hung. 94(4), 307–320 (2002)

    Article  MATH  Google Scholar 

  37. Bridson, M., Haefliger, A.: Metric Spaces of Non-positive Curvature. Springer, Berlin, Heidelberg, New York (1999)

    MATH  Google Scholar 

  38. Li, C., López, G., Martín-Márquez, V.: Iterative algorithms for nonexpansive mappings in Hadamard manifolds. Taiwan. J. Math. 14(2), 541–559 (2010)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, Zhejiang, 310032, P.R. China

    J. H. Wang

  2. Departamento de Análisis Matemático, Universidad de Sevilla, Apdo. 1160, 41080, Sevilla, Spain

    G. López & V. Martín-Márquez

  3. Department of Mathematics, Zhejiang University, Hangzhou, 310027, P.R. China

    C. Li

  4. Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

    C. Li

Authors
  1. J. H. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. López
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. Martín-Márquez
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. C. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. H. Wang.

Additional information

Communicated by J.-C. Yao.

G. López partially supported by Ministerio de Ciencia e Innovación, Grant MTM2009-110696-C02-01 and Junta de Andalucía, Grant FQM-127.

V. Martín-Márquez partially supported by Ministerio de Ciencia e Innovación, Grants MTM2009-110696-C02-01 and AP2005-1018, and Junta de Andalucía, Grant FQM-127.

C. Li partially supported by Ministerio de Ciencia e Innovación, Grant MTM2009-110696-C02-01, Spain; the National Natural Science Foundations of China (Grant No. 10731060).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, J.H., López, G., Martín-Márquez, V. et al. Monotone and Accretive Vector Fields on Riemannian Manifolds. J Optim Theory Appl 146, 691–708 (2010). https://doi.org/10.1007/s10957-010-9688-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-010-9688-z

Keywords

Search

Navigation