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On the Superlinear Convergence of Newton’s Method on Riemannian Manifolds

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Abstract

In this paper, we study Newton’s method for finding a singularity of a differentiable vector field defined on a Riemannian manifold. Under the assumption of invertibility of the covariant derivative of the vector field at its singularity, we show that Newton’s method is well defined in a suitable neighborhood of this singularity. Moreover, we show that the sequence generated by Newton’s method converges to the solution with superlinear rate.

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References

  1. Smith, S.T.: Optimization techniques on Riemannian manifolds, Hamiltonian and gradient flows, algorithms and control. In: Bloch, A. (ed.) Fields Institute Communication, vol. 3, pp. 113–136. American Mathematical Society, Providence, RI (1994)

  2. Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton, NJ (2008)

    Book  MATH  Google Scholar 

  3. Luenberger, D.G.: The gradient projection method along geodesics. Manag. Sci. 18, 620–631 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gabay, D.: Minimizing a differentiable function over a differential manifold. J. Optim. Theory Appl. 37(2), 177–219 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  5. Rapcsák, T.: Minimum problems on differentiable manifolds. Optimization 20(1), 3–13 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Udrişte, C.: Convex Functions and Optimization Methods on Riemannian Manifolds, Mathematics and its Applications, vol. 297. Kluwer Academic Publishers, Dordrecht (1994)

    Book  MATH  Google Scholar 

  7. Absil, P.A., Amodei, L., Meyer, G.: Two Newton methods on the manifold of fixed-rank matrices endowed with Riemannian quotient geometries. Comput. Stat. 29(3–4), 569–590 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Huang, W., Gallivan, K.A., Absil, P.A.: A Broyden class of quasi-Newton methods for Riemannian optimization. SIAM J. Optim. 25(3), 1660–1685 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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 

  10. Li, C., Wang, J.: Newton’s method for sections on Riemannian manifolds: generalized covariant \(\alpha \)-theory. J. Complex. 24(3), 423–451 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hu, Y., Li, C., Yang, X.: On convergence rates of linearized proximal algorithms for convex composite optimization with applications. SIAM J. Optim. 26(2), 1207–1235 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ring, W., Wirth, B.: Optimization methods on Riemannian manifolds and their application to shape space. SIAM J. Optim. 22(2), 596–627 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Manton, J.H.: A framework for generalising the Newton method and other iterative methods from Euclidean space to manifolds. Numer. Math. 129(1), 91–125 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bittencourt, T., Ferreira, O.: Kantorovichs theorem on Newtons method under majorant condition in Riemannian manifolds. J. Glob. Optim. (2016). doi:10.1007/s10898-016-0472

  15. Adler, R.L., Dedieu, J.P., Margulies, J.Y., Martens, M., Shub, M.: Newton’s method on Riemannian manifolds and a geometric model for the human spine. IMA J. Numer. Anal. 22(3), 359–390 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nash, J.: The imbedding problem for Riemannian manifolds. Ann. Math. 2(63), 20–63 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  17. Moser, J.: A new technique for the construction of solutions of nonlinear differential equations. Proc. Natl. Acad. Sci. USA 47, 1824–1831 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, C., Wang, J.H., Dedieu, J.P.: Smale’s point estimate theory for Newton’s method on Lie groups. J. Complex. 25(2), 128–151 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Schulz, V.H.: A Riemannian view on shape optimization. Found. Comput. Math. 14(3), 483–501 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang, J.H., Li, C.: Kantorovich’s theorems for Newton’s method for mappings and optimization problems on Lie groups. IMA J. Numer. Anal. 31(1), 322–347 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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  MATH  Google Scholar 

  23. 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  MATH  Google Scholar 

  24. Li, C., Wang, J.: Newton’s method on Riemannian manifolds: Smale’s point estimate theory under the \(\gamma \)-condition. IMA J. Numer. Anal. 26(2), 228–251 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ortega, J.M.: Numerical Analysis. A Second Course. Computer Science and Applied Mathematics. Academic Press, New York (1972)

    Google Scholar 

  26. 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  MATH  Google Scholar 

  27. do Carmo, M.P.: Riemannian Geometry. Mathematics: Theory & Applications. Birkhäuser, Boston (1992)

    Book  Google Scholar 

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

    Google Scholar 

  29. Dieudonné, J.: Foundations of Modern Analysis, vol. 10-I. Academic Press, New York (1969). Enlarged and Corrected Printing, Pure and Applied Mathematics

  30. Ferreira, O.P., Iusem, A.N., Németh, S.Z.: Concepts and techniques of optimization on the sphere. TOP 22(3), 1148–1170 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Rothaus, O.S.: Domains of positivity. Abh. Math. Sem. Univ. Hambg. 24, 189–235 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  32. Nesterov, Y.E., Todd, M.J.: On the Riemannian geometry defined by self-concordant barriers and interior-point methods. Found. Comput. Math. 2(4), 333–361 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lang, S.: Fundamentals of Differential Geometry, Graduate Texts in Mathematics, vol. 191. Springer, New York (1999)

    Book  Google Scholar 

  34. Milnor, J.: Morse theory. Based on lecture notes by M. Spivak and R. Wells. In: Annals of Mathematics Studies, No. 51. Princeton University Press, Princeton (1963)

  35. Kristály, A., Rădulescu, V.D., Varga, C.G.: Variational Principles in Mathematical Physics, Geometry, and Economics, Encyclopedia of Mathematics and its Applications, vol. 136. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

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Acknowledgements

The work was supported by FAPEG, UESB, and CNPq Grants 305158/2014-7 and 408151/2016-1.

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Authors and Affiliations

  1. Universidade Estadual do Sudoeste da Bahia, Vitória da Conquista, BA, 45083-900, Brazil

    Teles A. Fernandes

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

    Orizon P. Ferreira

  3. Universidade Federal do Paraná, Curitiba, PR, 81531-980, Brazil

    Jinyun Yuan

Authors
  1. Teles A. Fernandes
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  2. Orizon P. Ferreira
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  3. Jinyun Yuan
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Correspondence to Teles A. Fernandes.

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Communicated by Alexandru Kristály.

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Fernandes, T.A., Ferreira, O.P. & Yuan, J. On the Superlinear Convergence of Newton’s Method on Riemannian Manifolds. J Optim Theory Appl 173, 828–843 (2017). https://doi.org/10.1007/s10957-017-1107-2

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  • DOI: https://doi.org/10.1007/s10957-017-1107-2

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