Skip to main content
SpringerLink
Log in

A Riemannian derivative-free Polak–Ribiére–Polyak method for tangent vector field

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

This paper is concerned with the problem of finding a zero of a tangent vector field on a Riemannian manifold. We first reformulate the problem as an equivalent Riemannian optimization problem. Then, we propose a Riemannian derivative-free Polak–Ribiére–Polyak method for solving the Riemannian optimization problem, where a non-monotone line search is employed. The global convergence of the proposed method is established under some mild assumptions. To further improve the efficiency, we also provide a hybrid method, which combines the proposed geometric method with the Riemannian Newton method. Finally, some numerical experiments are reported to illustrate the efficiency of the proposed method.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

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

    Book  MATH  Google Scholar 

  2. Absil, P.-A., Ishteva, M., Lathauwer, L., van Huffel, S.: A geometric Newton method for Oja’s vector field. Neural Comput. 21, 1415–1433 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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, 359–390 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bento, G.C., Cruz Neto, J.X.: Finite termination of the proximal point method for convex functions on Hadamard manifolds. Optimization 63, 1281–1288 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bortoloti, M.A.A., Fernandes, T.A., Ferreira, O.P., Yuan, J.Y.: Damped Newton’s Method on Riemannian Manifolds. arXiv:1803.05126 (2018)

  6. Boumal, N., Mishra, B., Absil, P.-A., Sepulchre, R.: Manopt, a Matlab toolbox for optimization on manifolds. J. Mach. Learn. Res. 15, 1455–1459 (2014)

    MATH  Google Scholar 

  7. Cai, Y.F., Jia, Z.G., Bai, Z.J.: Perturbation analysis of an eigenvector-dependent nonlinear eigenvalue problem with applications, BIT. https://doi.org/10.1007/s10543-019-00765-4 (2019)

  8. Chen, H., Dai, X., Gong, X., He, L., Zhou, A.: Adaptive finite element approximations for Kohn-Sham models. Multiscale Model. Simul. 12, 1828–1869 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cheng, W., Li, D.: A derivative-free non-monotone line search and its applications to the spectral residual method. IMA J. Numer. Anal. 29, 814–825 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cheng, W., Xiao, Y., Hu, Q.J.: A family of derivative-free conjugate gradient methods for large-scale nonlinear systems of equations. J. Comput. Appl. Math. 224, 11–19 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cruz, W.L., Martínez, J., Raydan, M.: Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math. Comp. 75, 1429–448 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cruz, W.L., Raydan, M.: Nonmonotone spectral methods for large-scale nonlinear systems. Optim. Methods Softw. 18, 583–599 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R., Németh, S.Z.: Convex and monotone-transformable mathematical programming problems and a proximal-like point method. J. Global Optim. 35, 53–69 (2006)

    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. Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20, 303–353 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fang, X.W., Ni, Q.: A new derivative-free conjugate gradient method for large-scale nonlinear systems of equations. Bull. Aust. Math. Soc. 95, 500–511 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore and London (1996)

    MATH  Google Scholar 

  22. Helmke, U., Shayman, M.A.: Critical points of matrix least squares distance functions. Linear Algebra Appl. 215, 1–19 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jeuris, B.: Riemannian Optimization for Averaging Positive Definite Matrices, Ph.D. Dissertation, Department of Computer Science, KU Leuven (2015)

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

    Article  MathSciNet  MATH  Google Scholar 

  25. 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 

  26. Li, M.: A derivative-free PRP method for solving large-scale nonlinear systems of equations and its global convergence. Optim. Methods Softw. 29, 503–514 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Martin, R.M.: Electronic Structure: Basic Theory and Practical Methods. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  29. Ngo, T., Bellalij, M., Saad, Y.: The trace ratio optimization problem for dimensionality reduction. SIAM J. Matrix Anal. Appl. 31, 2950–2971 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Oja, E.: A simplified neuron model as a principal component analyzer. J. Math. Biol. 15, 267–273 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  31. Oja, E.: Neural networks, principal components, and subspaces. Neural Syst. 1, 61–68 (1989)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Saad, Y., Chelikowsky, J.R., Shontz, S.M.: Numerical methods for electronic structure calculations of materials. SIAM Rev. 52, 3–54 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sato, H., Iwai, T.: A new, globally convergent Riemannian conjugate gradient method. Optimization 64, 1011–1031 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tang, G.J., Huang, N.J.: An inexact proximal point algorithm for maximal monotone vector fields on Hadamard manifolds. Oper. Res. Lett. 41, 586–591 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, J.H., Li, C., Lopez, G., Yao, J.C.: Convergence analysis of inexact proximal point algorithms on Hadamard manifolds. J. Global Optim. 61, 553–573 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, J.H., Li, C., Lopez, G., Yao, J.C.: Proximal point algorithms on Hadamard manifolds: linear convergence and finite termination. SIAM J. Optim. 26, 2696–2729 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wang, J.H., López, G., Martín-Márquez, V., Li, C.: Monotone and accretive operators on Riemannian manifolds. J. Optim. Theory Appl. 146, 691–708 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Yao, T.T., Bai, Z.J., Zhao, Z., Ching, W.K.: A Riemannian Fletcher–Reeves conjugate gradient method for doubly stochastic inverse eigenvalue problems. SIAM J. Matrix Anal. Appl. 37, 215–234 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yao, T.T., Bai, Z.J., Jin, X.Q., Zhao, Z.: A geometric Gauss–Newton method for least squares inverse eigenvalue problems, BIT Numer Math. https://doi.org/10.1007/s10543-019-00798-9 (2020)

  41. Yu, G.H.: A derivative-free method for solving large-scale nonlinear systems of equations. J. Ind. Manag. Optim. 6, 149–160 (2010)

    MathSciNet  MATH  Google Scholar 

  42. Yu, G.H.: Nonmonotone spectral gradient-type methods for large-scale unconstrained optimization and nonlinear systems of equations. Pac. J. Optim. 7, 387–404 (2011)

    MathSciNet  MATH  Google Scholar 

  43. Zhang, L.H., Li, R.C.: Maximization of the sum of the trace ratio on the Stiefel manifold, I: theory. Sci. China Math. 57, 2495–2508 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhang, L.H., Li, R.C.: Maximization of the sum of the trace ratio on the Stiefel manifold, II: computation. Sci. China Math. 58, 1549–1566 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhao, Z., Bai, Z.J., Jin, X.Q.: A Riemannian Newton algorithm for nonlinear eigenvalue problems. SIAM J. Matrix Anal. Appl. 36, 752–774 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhao, Z., Jin, X.Q., Bai, Z.J.: A geometric nonlinear conjugate gradient method for stochastic inverse eigenvalue problems. SIAM J. Numer. Anal. 54, 2015–2035 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  47. Zhu, X.: A Riemannian conjugate gradient method for optimization on the Stiefel manifold. Comput. Optim. Appl. 67, 73–110 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the associate editor and the referees for their valuable comments.

Funding

This research is supported by the National Natural Science Foundation of China (No. 11701514, No. 11601112, and No. 11671337). Also, this research is supported by the Fundamental Research Funds for the Central Universities (No. 20720180008) and the research grants MYRG2016- 00077-FST, CPG2019-00030-FST, and MYRG2019-00042-FST from the University of Macau.

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Sciences, Zhejiang University of Science and Technology, Hangzhou, 310023, People’s Republic of China

    Teng-Teng Yao

  2. Department of Mathematics, School of Sciences, Hangzhou Dianzi University, Hangzhou, 310018, People’s Republic of China

    Zhi Zhao

  3. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing, Xiamen University, Xiamen, 361005, People’s Republic of China

    Zheng-Jian Bai

  4. Department of Mathematics, University of Macau, Macao, People’s Republic of China

    Xiao-Qing Jin

Authors
  1. Teng-Teng Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhi Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zheng-Jian Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiao-Qing Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zheng-Jian Bai.

Additional information

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

Yao, TT., Zhao, Z., Bai, ZJ. et al. A Riemannian derivative-free Polak–Ribiére–Polyak method for tangent vector field. Numer Algor 86, 325–355 (2021). https://doi.org/10.1007/s11075-020-00891-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-020-00891-z

Keywords

Mathematics Subject Classification (2010)

Search

Navigation