Skip to main content
SpringerLink
Log in

A Global Newton Method for the Nonsmooth Vector Fields on Riemannian Manifolds

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

Abstract

This paper proposes and analyzes a globalized version of the Newton method for finding a singularity of the nonsmooth vector fields. Basically, the new method combines a version of nonsmooth Newton method with a nonmonotone line search strategy. The global convergence analysis of the proposed method as well as results on its rate are established under mild assumptions. Finally, numerical experiments illustrating the practical advantages of the proposed scheme are reported.

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.

Similar content being viewed by others

References

  1. Argyros, I.K., Hilout, S.D.: Newton’s method for approximating zeros of vector fields on Riemannian manifolds. J. Appl. Math. Comput. 29(1–2), 417–427 (2009)

    Article  MathSciNet  Google Scholar 

  2. Azagra, D., Ferrera, J., López-Mesas, F.: Nonsmooth analysis and Hamilton–Jacobi equations on Riemannian manifolds. J. Funct. Anal. 220(2), 304–361 (2005)

    Article  MathSciNet  Google Scholar 

  3. Behling, R., Fischer, A., Herrich, M., Iusem, A., Ye, Y.: A Levenberg–Marquardt method with approximate projections. Comput. Optim. Appl. 59(1–2), 5–26 (2014)

    Article  MathSciNet  Google Scholar 

  4. Bello Cruz, J.Y., Ferreira, O.P., Prudente, L.F.: On the global convergence of the inexact semi-smooth Newton method for absolute value equation. Comput. Optim. Appl. 65(1), 93–108 (2016)

    Article  MathSciNet  Google Scholar 

  5. Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (2014)

    MATH  Google Scholar 

  6. Birgin, E.G., Martínez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10(4), 1196–1211 (2000)

    Article  MathSciNet  Google Scholar 

  7. Bittencourt, T., Ferreira, O.P.: Local convergence analysis of inexact Newton method with relative residual error tolerance under majorant condition in Riemannian manifolds. Appl. Math. Comput. 261, 28–38 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Bortoloti, M.A.D.A., Fernandes, T.A., Ferreira, O.P., Yuan, J.: Damped Newton’s method on Riemannian manifolds. J. Glob. Optim. 77(3), 643–660 (2020)

    Article  MathSciNet  Google Scholar 

  9. Canary, R.D., Epstein, D., Marden, A.: Fundamentals of hyperbolic geometry: selected expositions. London Mathematical Society Lecture Note Series, vol. 328. Cambridge University Press, Cambridge (2006)

  10. Clarke, F.H.: Optimization and nonsmooth analysis. Classics in Applied Mathematics, vol. 5, second edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1990)

  11. da Cruz Neto, J.X., de Lima, L.L., Oliveira, P.R.: Geodesic algorithms in Riemannian geometry. Balkan J. Geom. Appl. 3(2), 89–100 (1998)

    MathSciNet  MATH  Google Scholar 

  12. de Oliveira, F.R., Ferreira, O.P.: Newton method for finding a singularity of a special class of locally Lipschitz continuous vector fields on Riemannian manifolds. J. Optim. Theory Appl. 185(2), 522–539 (2020)

    Article  MathSciNet  Google Scholar 

  13. Dedieu, J.P., Priouret, P., Malajovich, G.: Newton’s method on Riemannian manifolds: convariant alpha theory. IMA J. Numer. Anal. 23(3), 395–419 (2003)

    Article  MathSciNet  Google Scholar 

  14. Dennis Jr., J.E., Schnabel, R.B.: Numerical methods for unconstrained optimization and nonlinear equations. Classics in Applied Mathematics, vol. 16. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1996). Corrected reprint of the 1983 original

  15. do Carmo, M.P.: Riemannian geometry. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA (1992). Translated from the second Portuguese edition by Francis Flaherty

  16. Facchinei, F., Soares, J.A.: A new merit function for nonlinear complementarity problems and a related algorithm. SIAM J. Optim. 7(1), 225–247 (1997)

    Article  MathSciNet  Google Scholar 

  17. Fan, J.: On the Levenberg-Marquardt methods for convex constrained nonlinear equations. J. Ind. Manag. Optim. 9(1), 227–241 (2013)

    MathSciNet  MATH  Google Scholar 

  18. Fernandes, T.A., Ferreira, O.P., Yuan, J.: On the superlinear convergence of Newton’s method on Riemannian manifolds. J. Optim. Theory Appl. 173(3), 828–843 (2017)

    Article  MathSciNet  Google Scholar 

  19. Ferreira, O.P.: Dini derivative and a characterization for Lipschitz and convex functions on Riemannian manifolds. Nonlinear Anal. 68(6), 1517–1528 (2008)

    Article  MathSciNet  Google Scholar 

  20. Ferreira, O.P., Silva, R.C.M.: Local convergence of Newton’s method under a majorant condition in Riemannian manifolds. IMA J. Numer. Anal. 32(4), 1696–1713 (2012)

    Article  MathSciNet  Google Scholar 

  21. Ferreira, O.P., Svaiter, B.F.: Kantorovich’s theorem on Newton’s method in Riemannian manifolds. J. Complex. 18(1), 304–329 (2002)

    Article  MathSciNet  Google Scholar 

  22. Ghahraei, E., Hosseini, S., Pouryayevali, M.R.: Pseudo-Jacobian and characterization of monotone vector fields on Riemannian manifolds. J. Convex Anal. 24(1), 149–168 (2017)

    MathSciNet  MATH  Google Scholar 

  23. Gonçalves, D.S., Gonçalves, M.L.N., Oliveira, F.R.: An inexact projected LM type algorithm for solving convex constrained nonlinear equations. J. Comput. Appl. Math. (2021). https://doi.org/10.1016/j.cam.2021.113421

  24. Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton’s method. SIAM J. Numer. Anal. 23(4), 707–716 (1986)

    Article  MathSciNet  Google Scholar 

  25. Grohs, P., Hosseini, S.: Nonsmooth trust region algorithms for locally Lipschitz functions on Riemannian manifolds. IMA J. Numer. Anal. 36(3), 1167–1192 (2016)

    Article  MathSciNet  Google Scholar 

  26. Hosseini, S., Pouryayevali, M.R.: Nonsmooth optimization techniques on Riemannian manifolds. J. Optim. Theory Appl. 158(2), 328–342 (2013)

    Article  MathSciNet  Google Scholar 

  27. Hosseini, S., Uschmajew, A.: A Riemannian gradient sampling algorithm for nonsmooth optimization on manifolds. SIAM J. Optim. 27(1), 173–189 (2017)

    Article  MathSciNet  Google Scholar 

  28. Hosseini, S., Huang, W., Yousefpour, R.: Line search algorithms for locally Lipschitz functions on Riemannian manifolds. SIAM J. Optim. 28(1), 596–619 (2018)

    Article  MathSciNet  Google Scholar 

  29. Iannazzo, B., Porcelli, M.: The Riemannian Barzilai–Borwein method with nonmonotone line search and the matrix geometric mean computation. IMA J. Numer. Anal. 38(1), 495–517 (2018)

    Article  MathSciNet  Google Scholar 

  30. Lang, S.: Differential and Riemannian manifolds. Graduate Texts in Mathematics, vol. 160, third edn. Springer-Verlag, New York (1995)

  31. Ledyaev, Y.S., Zhu, Q.J.: Nonsmooth analysis on smooth manifolds. Trans. Am. Math. Soc. 359(8), 3687–3732 (2007)

    Article  MathSciNet  Google Scholar 

  32. Li, C., Wang, J.: Convergence of the newton method and uniqueness of zeros of vector fields on riemannian manifolds. Sci. China Ser. A Math. 48(11), 1465–1478 (2005)

    Article  MathSciNet  Google Scholar 

  33. Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3(1), 101–108 (2009)

    Article  MathSciNet  Google Scholar 

  34. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  35. Rampazzo, F., Sussmann, H.J.: Commutators of flow maps of nonsmooth vector fields. J. Differ. Equ. 232(1), 134–175 (2007)

    Article  MathSciNet  Google Scholar 

  36. Raydan, M.: The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J. Optim. 7(1), 26–33 (1997)

    Article  MathSciNet  Google Scholar 

  37. Sakai, T.: Riemannian geometry. Translations of Mathematical Monographs, vol. 149. American Mathematical Society, Providence, RI (1996). Translated from the 1992 Japanese original by the author

  38. Tu, L.W.: An Introduction to Manifolds, second edn. Universitext. Springer, New York (2011)

  39. Wang, J.H.: Convergence of Newton’s method for sections on Riemannian manifolds. J. Optim. Theory Appl. 148(1), 125–145 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IME, Universidade Federal de Goiás, Goiânia, GO, 74001-970, Brazil

    Fabiana R. de Oliveira & Fabrícia R. Oliveira

Authors
  1. Fabiana R. de Oliveira
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fabrícia R. Oliveira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fabiana R. de Oliveira.

Additional information

Communicated by Sándor Zoltán Németh.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de Oliveira, F.R., Oliveira, F.R. A Global Newton Method for the Nonsmooth Vector Fields on Riemannian Manifolds. J Optim Theory Appl 190, 259–273 (2021). https://doi.org/10.1007/s10957-021-01881-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-021-01881-4

Keywords

Mathematics Subject Classification

Search

Navigation