Journal of Optimization Theory and Applications

, Volume 180, Issue 3, pp 993–1010 | Cite as

On the Local and Superlinear Convergence of a Secant Modified Linear-Programming-Newton Method

  • María de los Ángeles MartínezEmail author
  • Damián Fernández


We present a superlinearly convergent method to solve a constrained system of nonlinear equations. The proposed procedure is an adaptation of the linear-programming-Newton method replacing the first-order information with a secant update. Thus, under mild assumptions, the method is able to find possible nonisolated solutions without computing any derivative and achieving a local superlinear rate of convergence. In addition to the convergence analysis, some numerical examples are presented in order to show the fulfillment of the expected rate of convergence.


Constrained nonlinear system of equations Nonisolated solutions Quasi-Newton method Local superlinear convergence 

Mathematics Subject Classification

90C30 65K05 



This work was partially supported by FONCyT Grant PICT 2014-2534 and CONICET Grant PIP 112-201101-00050.


  1. 1.
    Facchinei, F., Fischer, A., Herrich, M.: An LP-Newton method: nonsmooth equations, KKT systems, and nonisolated solutions. Math. Program. 146(1–2), 1–36 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Martínez, M.D.L.Á., Fernández, D.: A quasi-Newton modified LP-Newton method. In: Optimization Methods and Software, pp. 1–16 (2017)Google Scholar
  3. 3.
    Dennis Jr., J.E., Schnabel, R.B.: Least change secant updates for quasi-Newton methods. Siam Rev. 21(4), 443–459 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Martınez, J.M.: Practical quasi-Newton methods for solving nonlinear systems. J. Comput. Appl. Math. 124(1), 97–121 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19(2), 400–408 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Izmailov, A.F., Solodov, M.V.: Newton-Type Methods for Optimization and Variational Problems. Springer, Berlin (2014)CrossRefzbMATHGoogle Scholar
  7. 7.
    Kanzow, C.: An active set-type Newton method for constrained nonlinear systems. Complement. Appl. Algorithms Ext. 50, 179–200 (2001)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kozakevich, D.N., Martinez, J.M., Santos, S.A.: Solving nonlinear systems of equations with simple constraints. Comput. Appl. Math. 16(3), 215–235 (1997)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ulbrich, M.: Nonmonotone trust-region methods for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems. SIAM J. Optim. 11(4), 889–917 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Yamashita, N., Fukushima, M.: On the rate of convergence of the Levenberg–Marquardt method. In: Alefeld, G., Chen, X. (eds.) Topics in Numerical Analysis. Computing Supplementa, vol. 15, pp. 239–249. Springer, Vienna (2001)CrossRefGoogle Scholar
  11. 11.
    Fan, J.Y., Yuan, Y.X.: On the quadratic convergence of the Levenberg–Marquardt method without nonsingularity assumption. Computing 74(1), 23–39 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fischer, A.: Local behavior of an iterative framework for generalized equations with nonisolated solutions. Math. Program. 94(1), 91–124 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fischer, A., Herrich, M., Izmailov, A.F., Solodov, M.V.: A globally convergent LP-Newton method. SIAM J. Optim. 26(4), 2012–2033 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Izmailov, A.F., Solodov, M.V.: On error bounds and Newton-type methods for generalized nash equilibrium problems. Comput. Optim. Appl. 59(1–2), 201–218 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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(2), 425–459 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bonnans, J.F.: Local analysis of Newton-type methods for variational inequalities and nonlinear programming. Appl. Math. Optim. 29(2), 161–186 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fan, J.: Convergence rate of the trust region method for nonlinear equations under local error bound condition. Comput. Optim. Appl. 34(2), 215–227 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.CIEM, FaMAF, CONICETUniversidad Nacional de CórdobaCórdobaArgentina

Personalised recommendations