A global hybrid derivative-free method for high-dimensional systems of nonlinear equations

  • Rodolfo G. BegiatoEmail author
  • Ana L. Custódio
  • Márcia A. Gomes-Ruggiero


This work concerns the numerical solution of high-dimensional systems of nonlinear equations, when derivatives are not available for use, but assuming that all functions defining the problem are continuously differentiable. A hybrid approach is taken, based on a derivative-free iterative method, organized in two phases. The first phase is defined by derivative-free versions of a fixed-point method that employs spectral parameters to define the steplength along the residual direction. The second phase consists on a matrix-free inexact Newton method that employs the Generalized Minimal Residual algorithm to solve the linear system that computes the search direction. This second phase will only take place if the first one fails to find a better point after a predefined number of reductions in the step size. In all stages, the criterion to accept a new point considers a nonmonotone decrease condition upon a merit function. Convergence results are established and the numerical performance is assessed through experiments in a set of problems collected from the literature. Both the theoretical and the experimental analysis support the feasibility of the proposed hybrid strategy.


Nonlinear systems of equations Derivative-free optimization methods Spectral residual Inexact Newton Nonmonotone line search 



We are thankful to Sandra Augusta Santos, from University of Campinas, Brazil, for suggestions and valuable discussions on the subject, which greatly helped us to improve the paper. We also acknowledge the suggestions of two anonymous referees, which greatly contributed for the paper organization.


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Authors and Affiliations

  1. 1.DAMATUTFPRCuritibaBrazil
  2. 2.Department of MathematicsFCT-UNL-CMACaparicaPortugal
  3. 3.Departamento de Matemática Aplicada, IMECCUNICAMPCampinasBrazil

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