Skip to main content

Advertisement

Springer Nature Link
Log in

Local Attractors of Newton-Type Methods for Constrained Equations and Complementarity Problems with Nonisolated Solutions

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

Abstract

For constrained equations with nonisolated solutions, we show that if the equation mapping is 2-regular at a given solution with respect to a direction in the null space of the Jacobian, and this direction is interior feasible, then there is an associated domain of starting points from which a family of Newton-type methods is well defined and necessarily converges to this specific solution (despite degeneracy, and despite that there are other solutions nearby). We note that unlike the common settings of convergence analyses, our assumptions subsume that a local Lipschitzian error bound does not hold for the solution in question. Our results apply to constrained and projected variants of the Gauss–Newton, Levenberg–Marquardt, and LP-Newton methods. Applications to smooth and piecewise smooth reformulations of complementarity problems are also discussed.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Facchinei, F., Fischer, A., Herrich, M.: An LP-Newton method: nonsmooth equations, KKT systems, and nonisolated solutions. Math. Program. 146, 1–36 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Fischer, A., Herrich, M., Izmailov, A.F., Solodov, M.V.: Convergence conditions for Newton-type methods applied to complementarity systems with nonisolated solutions. Comput. Optim. Appl. 63, 425–459 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Izmailov, A.F., Solodov, M.V.: On attraction of Newton-type iterates to multipliers violating second-order sufficiency conditions. Math. Program. 117, 271–304 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Izmailov, A.F., Solodov, M.V.: On attraction of linearly constrained Lagrangian methods and of stabilized and quasi-Newton SQP methods to critical multipliers. Math. Program. 126, 231–257 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Izmailov, A.F., Solodov, M.V.: Newton-Type Methods for Optimization and Variational Problems. Springer Series in Operations Research and Financial Engineering. Springer International Publishing, Cham (2014)

    Google Scholar 

  6. Izmailov, A.F., Kurennoy, A.S., Solodov, M.V.: Critical solutions of nonlinear equations: stability issues. Math. Program. 168, 475–507 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Izmailov, A.F., Kurennoy, A.S., Solodov, M.V.: Critical solutions of nonlinear equations: local attraction for Newton-type methods. Math. Program. 167, 355–379 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kanzow, C., Yamashita, N., Fukushima, M.: Levenberg–Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints. J. Comput. Appl. Math. 172, 375–397 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. SIAM, Philadelphia (2009)

    Book  MATH  Google Scholar 

  10. Griewank, A.: Starlike domains of convergence for Newton’s method at singularities. Numer. Math. 35, 95–111 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  11. Oberlin, C., Wright, S.J.: An accelerated Newton method for equations with semismooth Jacobians and nonlinear complementarity problems. Math. Program. 117, 355–386 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Facchinei, F., Fischer, A., Kanzow, C., Peng, J.-M.: A simply constrained optimization reformulation of KKT systems arising from variational inequalities. Appl. Math. Optim. 40, 19–37 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhang, J.-L.: On the convergence properties of the Levenberg–Marquardt method. Optimization 52, 739–756 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Behling, R., Fischer, A.: A unified local convergence analysis of inexact constrained Levenberg–Marquardt methods. Optim. Lett. 6, 927–940 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Behling, R.: The method and the trajectory of Levenberg–Marquardt. Ph.D. thesis. Preprint C130/2011. IMPA – Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro (2011)

  16. Fischer, A., Shukla, P.K., Wang, M.: On the inexactness level of robust Levenberg–Marquardt methods. Optimization 59, 273–287 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Behling, R., Fischer, A., Haeser, G., Ramos, A., Schönefeld, K.: On the constrained error bound condition and the projected Levenberg–Marquardt method. Optimization 66, 1397–1411 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Arutyunov, A.V.: Optimality Conditions: Abnormal and Degenerate Problems. Kluwer Academic Publishers, Dordrecht (2000)

    Book  MATH  Google Scholar 

  20. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  21. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)

    MATH  Google Scholar 

  22. Arutyunov, A.V., Izmailov, A.F.: Stability of possibly nonisolated solutions of constrained equations, with applications to complementarity and equilibrium problems. Set-Valued Var. Anal. https://doi.org/10.1007/s11228-017-0459-y

Download references

Acknowledgements

Research of the first author is supported in part by the Volkswagen Foundation. Research of the second author is supported by the Russian Science Foundation Grant 17-11-01168. The third author is supported in part by CNPq Grant 303724/2015-3 and by FAPERJ Grant 203.052/2016.

Author information

Authors and Affiliations

  1. Faculty of Mathematics, Technische Universität Dresden, 01062, Dresden, Germany

    Andreas Fischer

  2. VMK Faculty, OR Department, Lomonosov Moscow State University, MSU, Uchebniy Korpus 2, Leninskiye Gory, Moscow, Russia, 119991

    Alexey F. Izmailov

  3. RUDN University, Miklukho-Maklaya Str. 6, Moscow, Russia, 117198

    Alexey F. Izmailov

  4. IMPA – Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, 22460-320, Brazil

    Mikhail V. Solodov

Authors
  1. Andreas Fischer
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Alexey F. Izmailov
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Mikhail V. Solodov
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Mikhail V. Solodov.

Additional information

Dedicated to Professor Aram Arutyunov on the occasion of his 60th birthday.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fischer, A., Izmailov, A.F. & Solodov, M.V. Local Attractors of Newton-Type Methods for Constrained Equations and Complementarity Problems with Nonisolated Solutions. J Optim Theory Appl 180, 140–169 (2019). https://doi.org/10.1007/s10957-018-1297-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-018-1297-2

Keywords

Mathematics Subject Classification