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Journal of Optimization Theory and Applications

, Volume 180, Issue 1, pp 140–169 | Cite as

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

  • Andreas Fischer
  • Alexey F. Izmailov
  • Mikhail V. SolodovEmail author
Article
  • 95 Downloads

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.

Keywords

Constrained equation Complementarity problem Nonisolated solution 2-Regularity Newton-type method Levenberg–Marquardt method LP-Newton method Piecewise Newton method 

Mathematics Subject Classification

47J05 90C33 65K15 

Notes

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.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of MathematicsTechnische Universität DresdenDresdenGermany
  2. 2.VMK Faculty, OR DepartmentLomonosov Moscow State University, MSUMoscowRussia
  3. 3.RUDN UniversityMoscowRussia
  4. 4.IMPA – Instituto de Matemática Pura e AplicadaRio de JaneiroBrazil

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