Skip to main content
SpringerLink
Log in

Convergence conditions for Newton-type methods applied to complementarity systems with nonisolated solutions

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We consider a class of Newton-type methods that are designed for the difficult case when solutions need not be isolated, and the equation mapping need not be differentiable at the solutions. We show that the only structural assumption needed for rapid local convergence of those algorithms applied to PC\(^1\)-equations is the piecewise error bound, i.e., a local error bound holding for the branches of the solution set resulting from partitions of the bi-active complementarity indices. The latter error bound is implied by various piecewise constraint qualifications, including relatively weak ones. We apply our results to KKT systems arising from optimization or variational problems, and from generalized Nash equilibrium problems. In the first case, we show convergence if the dual part of the solution is a noncritical Lagrange multiplier, and in the second case convergence follows under a relaxed constant rank condition. In both cases, previously available results are improved.

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

Similar content being viewed by others

References

  1. Andreani, R., Haeser, G., Schuverdt, M.L., Silva, P.J.S.: A relaxed constant positive linear dependence constraint qualification and applications. Math. Program. 135, 255–273 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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 

  3. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  MATH  Google Scholar 

  4. Dreves, A.: Improved error bound and a hybrid method for generalized Nash equilibrium problems. Comput. Optim. Appl. (2014). doi:10.1007/s10589-014-9699-z

  5. Dreves, A., Facchinei, F., Fischer, A., Herrich, M.: A new error bound result for generalized Nash equilibrium problems and its algorithmic application. Comput. Optim. Appl. 59, 63–84 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Facchinei, F., Fischer, A., Herrich, M.: A family of Newton methods for nonsmooth constrained systems with nonisolated solutions. Math. Methods Oper. Res. 77, 433–443 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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 

  8. Facchinei, F., Fischer, A., Piccialli, V.: Generalized Nash equilibrium problems and Newton methods. Math. Program. 117, 163–194 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. Ann. Oper. Res. 175, 177–211 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  11. Fernández, D., Solodov, M.: Stabilized sequential quadratic programming for optimization and a stabilized Newton-type method for variational problems. Math. Program. 125, 47–73 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fischer, A.: Modified Wilson’s method for nonlinear programs with nonunique multipliers. Math. Oper. Res. 24, 699–727 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fischer, A., Herrich, M., Schönefeld, K.: Generalized Nash equilibrium problems—recent advances and challenges. Pesquisa Operacional 34, 521–558 (2014)

    Article  Google Scholar 

  14. Hager, W.W.: Lipschitz continuity for constrained processes. SIAM J. Control Optim. 17, 321–338 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  15. Izmailov, A.F., Kurennoy, A.S., Solodov, M.V.: A note on upper Lipschitz stability, error bounds, and critical multipliers for Lipschitz-continuous KKT systems. Math. Program. 142, 591–604 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Izmailov, A.F., Kurennoy, A.S., Solodov, M.V.: The Josephy-Newton method for semismooth generalized equations and semismooth SQP for optimization. Set Valued Var. Anal. 21, 17–45 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Izmailov, A.F., Solodov, M.V.: Karush–Kuhn–Tucker systems: regularity conditions, error bounds and a class of Newton-type methods. Math. Program. 95, 631–650 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Izmailov, A.F., Solodov, M.V.: Newton-type methods for optimization problems without constraint qualifications. SIAM J. Optim. 15, 210–228 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Izmailov, A.F., Solodov, M.V.: Stabilized SQP revisited. Math. Program. 133, 93–120 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Izmailov, A.F., Solodov, M.V.: On error bounds and Newton-type methods for generalized Nash equilibrium problems. Comput. Optim. Appl. 59, 201–218 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. 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, Switzerland (2014)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Kojima, M., Shindo, S.: Extensions of Newton and quasi-Newton methods to systems of PC\(^1\) equations. J. Oper. Res. Soc. Jpn. 29, 352–374 (1986)

    MathSciNet  MATH  Google Scholar 

  24. Kyparisis, J.: Sensitivity analysis for nonlinear programs and variational inequalities with nonunique multipliers. Math. Oper. Res. 15, 286–298 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lu, S.: Relation between the constant rank and the relaxed constant rank constraint qualifications. Optimization 61, 555–666 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mangasarian, O.L., Fromovitz, S.: The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. J. Math. Anal. Appl. 17, 37–47 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  27. Minchenko, L., Stakhovski, S.: On relaxed constant rank regularity condition in mathematical programming. Optimization 60, 429–440 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Robinson, S.M.: Stability theory for systems of inequalities, part II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13, 497–513 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wright, S.J.: An algorithm for degenerate nonlinear programming with rapid local convergence. SIAM J. Optim. 15, 673–696 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Research of the Alexey F. Izmailov and Mikhail V. Solodov is supported in part by the Russian Foundation for Basic Research Grant 14-01-00113, by the Russian Science Foundation Grant 15-11-10021, by CNPq Grants 302637/2011-7 and PVE 401119/2014-9, by PRONEX–Optimization, and by FAPERJ.

Author information

Authors and Affiliations

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

    Andreas Fischer & Markus Herrich

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

    Alexey F. Izmailov

  3. IMPA – Instituto Nacional 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 in PubMed Google Scholar

  2. Markus Herrich
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexey F. Izmailov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mikhail V. Solodov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andreas Fischer.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fischer, A., Herrich, M., Izmailov, A.F. et al. Convergence conditions for Newton-type methods applied to complementarity systems with nonisolated solutions. Comput Optim Appl 63, 425–459 (2016). https://doi.org/10.1007/s10589-015-9782-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-015-9782-0

Keywords

Search

Navigation