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

  • María de los Ángeles Martínez
  • 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Martınez, J.M.: Practical quasi-Newton methods for solving nonlinear systems. J. Comput. Appl. Math. 124(1), 97–121 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19(2), 400–408 (1982)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Izmailov, A.F., Solodov, M.V.: Newton-Type Methods for Optimization and Variational Problems. Springer, Berlin (2014)CrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fischer, A.: Local behavior of an iterative framework for generalized equations with nonisolated solutions. Math. Program. 94(1), 91–124 (2002)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

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

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