nonlinear systems of equations

Secant acceleration of sequential residual methods for solving large-scale nonlinear systems of equations

  • Ernesto G. Birgin
  • José Mario Martínez
    • Categories Nonlinear Systems and Least-Squares Tags , , ,

    Sequential Residual Methods try to solve nonlinear systems of equations $F(x)=0$ by iteratively updating the current approximate solution along a residual-related direction. Therefore, memory requirements are minimal and, consequently, these methods are attractive for solving large-scale nonlinear systems. However, the convergence of these algorithms may be slow in critical cases; therefore, acceleration procedures are welcome. … Read more

    Approximate norm descent methods for constrained nonlinear systems

  • Benedetta Morini
  • Margherita Porcelli
  • Philippe L. Toint
    • Categories Bound-constrained Optimization, Nonlinear Systems and Least-Squares Tags , , ,

    We address the solution of convex-constrained nonlinear systems of equations where the Jacobian matrix is unavailable or its computation/storage is burdensome. In order to efficiently solve such problems, we propose a new class of algorithms which are “derivative-free” both in the computation of the search direction and in the selection of the steplength. Search directions … Read more

    Solving systems of nonlinear equations with continuous GRASP

  • Michael J. Hirsch
  • Panos M. Pardalos
  • Mauricio G.C. Resende
    • Categories Global Optimization, Meta Heuristics, Stochastic Approaches Tags , , , , , , , ,

    A method for finding all roots of a system of nonlinear equations is described. Our method makes use of C-GRASP, a recently proposed continuous global optimization heuristic. Given a nonlinear system, we solve a corresponding adaptively modified global optimization problem multiple times, each time using C-GRASP, with areas of repulsion around roots that have already … Read more

    Componentwise fast convergence in the solution of full-rank systems of nonlinear equations

  • Nicholas I. M. Gould
  • Dominique Orban
  • Annick Sartenaer
  • Philippe L. Toint
    • Categories Constrained Nonlinear Optimization, Nonlinear Systems and Least-Squares Tags , ,

    The asymptotic convergence of parameterized variants of Newton’s method for the solution of nonlinear systems of equations is considered. The original system is perturbed by a term involving the variables and a scalar parameter which is driven to zero as the iteration proceeds. The exact local solutions to the perturbed systems then form a differentiable … Read more

    On the Convergence of Newton Iterations to Non-Stationary Points

  • Richard H. Byrd
  • Marcelo Marazzi
  • Jorge Nocedal
    • Categories Nonlinear Optimization, Nonlinear Systems and Least-Squares, Unconstrained Optimization Tags , , ,

    We study conditions under which line search Newton methods for nonlinear systems of equations and optimization fail due to the presence of singular non-stationary points. These points are not solutions of the problem and are characterized by the fact that Jacobian or Hessian matrices are singular. It is shown that, for systems of nonlinear equations, … Read more