Skip to main content
SpringerLink
Log in

A Nonmonotone Trust Region Method for Unconstrained Optimization Problems on Riemannian Manifolds

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

Abstract

We propose a nonmonotone trust region method for unconstrained optimization problems on Riemannian manifolds. Global convergence to the first-order stationary points is proved under some reasonable conditions. We also establish local R-linear, super-linear and quadratic convergence rates. Preliminary experiments show that the algorithm is efficient.

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

Similar content being viewed by others

References

  1. Levenberg, K.: A method for the solution of certain non-linear problems in least squares. Quart. Appl. Math. 2, 164–168 (1994)

    Article  MathSciNet  Google Scholar 

  2. Ahookhosh, M., Amini, K.: A nonmonotone trust region method with adaptive radius for unconstrained optimization problems. Comput. Math. Appl. 60, 411–422 (2010)

    Article  MathSciNet  Google Scholar 

  3. Amini, K., Ahookhosh, M.: A hybrid of adjustable trust-region and nonmonotone algorithms for unconstrained optimization. Appl. Math. Model. 38, 2601–2612 (2014)

    Article  MathSciNet  Google Scholar 

  4. Shi, Z.J., Guo, J.H.: A new trust region methods for unconstrained optimization. Comput. Math. Appl. 213, 509–520 (2008)

    Article  MathSciNet  Google Scholar 

  5. Sorensen, D.C.: Newton’s method with a model trust region modification. SIAM J. Numer. Anal. 19, 409–426 (1982)

    Article  MathSciNet  Google Scholar 

  6. Sorensen, D.C.: Trust region methods for unconstrained minimization. CONF-8107106-2. Argonne National Lab., IL (USA) (1981)

  7. Yuan, Y.X.: Recent advances in trust region algorithms. Math. Program. 151, 249–281 (2015)

    Article  MathSciNet  Google Scholar 

  8. Powell, M.J.D.: On the global convergence of trust region algorithms for unconstrained minimization. Math. Program. 29, 297–303 (1984)

    Article  MathSciNet  Google Scholar 

  9. Zhang, X.S., Zhang, J.L., Liao, L.Z.: An nonmonotone adaptive trust region method and its convergence. Int. Comput. Math. Appl. 45, 1469–1477 (2003)

    Article  MathSciNet  Google Scholar 

  10. Zhang, H.C., Hager, W.W.: A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Optim. 14(4), 1043–1056 (2004)

    Article  MathSciNet  Google Scholar 

  11. Mo, J., Liu, C., Yan, S.: A nonmonotone trust region method based on nonincreasing technique of weighted average of the successive function value. J. Comput. Appl. Math. 209, 97–108 (2007)

    Article  MathSciNet  Google Scholar 

  12. Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. SIAM, Philadelphia (2000)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Grippo, L., Lampariello, F., Lucidi, S.: A truncated Newton method with nonmonotone line search for unconstrained optimization. J. Optim. Theory Appl. 60(3), 401–419 (1989)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. 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  Google Scholar 

  18. Huang, W.: Optimization Algorithms on Riemannian Manifolds with Applications. Ph.D. Thesis, Department of Mathematics, Florida State University, Tallahassee, FL (2013)

  19. Li, X.B., Huang, N.J., Ansari, Q.H., Yao, J.C.: Convergence rate of descent method with new inexact line-search on Riemannian manifolds. J. Optim. Theory Appl. 180, 830–854 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Absil, P.A., Baker, C.G., Gallivan, K.A.: Trust-region methods on Riemannian manifolds. Found. Comput. Math. 7, 303–330 (2007)

    Article  MathSciNet  Google Scholar 

  22. Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Math., vol. 33, Am. Math. Soc., Providence, RI (2001)

  23. Chavel, I.: Riemannian Geometry: A Modern Introduction. Cambridge University Press, London (1993)

    MATH  Google Scholar 

  24. Klingernerg, W.: A Course in Differential Geometry. Springer, Berlin (1978)

    Book  Google Scholar 

  25. Boumal, N.: An Introduction to Optimization on Smooth Manifolds (2020)

  26. Sakai, T.: Riemannian Geometry, Translations of Mathematical Monograph. Amer. Math. Soc., Providence, RI (1996)

  27. Udrişte, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Kluwer Academic Publishers, Dordrecht (1994)

    Book  Google Scholar 

  28. Theis, F.J., Cason, T.P., Absil, P.A.: Soft dimension reduction for ICA by joint diagonalization on the Stiefel manifold. In: Adali, T., Jutten, C., Romano, J.M.T., Barros, A.K. (eds.) International Conference on Independent Component Analysis and Signal Separation. Lecture Notes in Computer Science, vol. 5441, pp. 354–361. Springer, Berlin (2009)

    Chapter  Google Scholar 

  29. Huang, W., Absil, P.A., Gallivan, K.A.: A Riemannian symmetric rank-one trust-region method. Math. Program Ser. A. 150, 179–216 (2015)

    Article  MathSciNet  Google Scholar 

  30. Sato, H.: Riemannian Newton-type method for joint diagonalization on the Stiefel manifold with application to independent component analysis. Optimization 66(12), 2211–2231 (2017)

    Article  MathSciNet  Google Scholar 

  31. Boumal, N., Absil, P.A.: RTRMC: a Riemannian trust-region method for low-rank matrix completion. In: Annual Conference on Neural Information Processing Systems 2011: Advances in Neural Information Processing Systems 24, pp. 406–414 (2011)

  32. Vandereycken, B.: Low-rank matrix completion by Riemannian optimization. SIAM J. Optim. 23, 1214–1236 (2013)

    Article  MathSciNet  Google Scholar 

  33. Jain, P., Netrapalli, P., Sanghavi, S.: Low-rank matrix completion using alternating minimization. In: STOC ’13: Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing 45, pp. 665-674 (2013)

  34. Kasai, H., Mishra, B.: Inexact trust-region algorithms on Riemannian manifolds. In: Annual Conference on Neural Information Processing Systems 2018: Advances in Neural Information Processing Systems 31, pp. 4249–4260 (2018)

  35. Boumal, N., Mishra, B., Absil, P.A., Sepulchre, R.: Manopt, a Matlab Toolbox for Optimization on Manifolds. J. Mach. Learn. Res. 15(1), 1455–1459 (2014)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the editors and referees for their constructive comments. X. Li was supported by the Young Scientists Fund of the National Natural Science Foundation of China (11901485). X. Wang was supported by the Natural Sciences and Engineering Research Council of Canada. M. Krishan Lal was supported by the UBC-SERB PhD fellowship.

Author information

Authors and Affiliations

  1. School of Sciences and Institute for Artificial Intelligence, Southwest Petroleum University, Chengdu, People’s Republic of China

    Xiaobo Li

  2. University of British Columbia Okanagan, Kelowna, Canada

    Xianfu Wang & Manish Krishan Lal

Authors
  1. Xiaobo Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xianfu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Manish Krishan Lal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xianfu Wang.

Additional information

Communicated by Alexandru Kristály.

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

Li, X., Wang, X. & Krishan Lal, M. A Nonmonotone Trust Region Method for Unconstrained Optimization Problems on Riemannian Manifolds. J Optim Theory Appl 188, 547–570 (2021). https://doi.org/10.1007/s10957-020-01796-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-020-01796-6

Keywords

Mathematics Subject Classification

Search

Navigation