A perturbed version of an inexact generalized Newton method for solving nonsmooth equations

Abstract

In this paper, we present the combination of the inexact Newton method and the generalized Newton method for solving nonsmooth equations F(x) = 0, characterizing the local convergence in terms of the perturbations and residuals. We assume that both iteration matrices taken from the B-differential and vectors F(x (k) ) are perturbed at each step. Some results are motivated by the approach of Cȃtinaş regarding to smooth equations. We study the conditions, which determine admissible magnitude of perturbations to preserve the convergence of method. Finally, the utility of these results is considered based on some variant of the perturbed inexact generalized Newton method for solving some general optimization problems.

Hybrid Second Order Method for Orthogonal Projection onto Parametric Curve in n-Dimensional Euclidean Space

“…We test many examples for arbitrary initial value, arbitrary test point and arbitrary parametric curve and find that our method remains more robust to converge than the H-H-H method. If the first order derivative f (t m ) of the iterative Equation (16) develops into 0, i.e., f (t m ) = 0 about some non-negative integer m, we use a perturbed method to solve the special problem, which adopts the idea in [23,45]. Namely, the function f (t m ) = 0 could be increased by a very small positive number ε, i.e., f (t m ) = f (t m ) + ε, and then the iteration by Equation (16) is continued in order to calculate the parameter value.…”

Inexact Newton method with feasible inexact projections for solving constrained smooth and nonsmooth equations

“…where η k ∈ [0, 1) is the relative residual error tolerance and V k is an element of the Clarke generalized Jacobian of f at x k (for the definition of the Clarke generalized Jacobian, see [9]). More versions of inexact Newton-type methods for solving (1) include, but are not limited to, those in [7,8,16,32,34,35,36]. Our aim in this paper is to study the inexact Newton method (2) with feasible inexact projections (inexact Newton-InexP method) for solving smooth and nonsmooth equations subject to a set of constraints, i.e., to solve the following constrained equation: Find x ∈ R n such that…”