A nonmonotone semismooth inexact Newton method

Abstract

In this work we propose a variant of the inexact Newton method for the solution of semismooth nonlinear systems of equations. We introduce a nonmonotone scheme, which couples the inexact features with the nonmonotone strategies. For the nonmonotone scheme, we present the convergence theorems. Finally, we show how we can apply these strategies in the variational inequalities context and we present some numerical examples.

Inexact quasi-Newton global convergent method for solving constrained nonsmooth equations

“…So, F x (k) = 0 for all k = 0, 1, 2, .... We may assume that α k < 1 for all k ∈ K 2 without loss of generality. By (5) and (4) we have that, for all k ∈ K 2…”

“…The smooth nonmonotone inexact Newton method was proposed by Bonettini in [4]. Bonettini and Tinti in [5] modified the general inexact Newton algorithm in a nonmonotone way for a semismooth equations. Our approach is similar as both versions presented in [4] (smooth) and [5] (semismooth).…”

“…Bonettini and Tinti in [5] modified the general inexact Newton algorithm in a nonmonotone way for a semismooth equations. Our approach is similar as both versions presented in [4] (smooth) and [5] (semismooth). In these papers the authors are mainly concerned with convergence of inexact Newton method for solving unconstrained semismooth equations while we study the inexact quasi-Newton method for the Lipschitz continuous equations with simple constraint.…”

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

“…Some substantial extension of method with the B-differential was also given in [35], where additionally a globally convergent hybrid method with Armijo line search was presented. In turn, Bonettini and Tinti [3] proposed a nonmonotone variant of the inexact generalized Newton method with backtracking strategy for solving semismooth equations. Moreover, in [34] we introduce some parameterized version of the method described by (3) and (4) with the fixed forcing terms η k for solving constrained nonsmooth equations.…”

“…[1,2,14]. Furthermore, the nonsmooth versions of the inexact generalized Newton method were considered among the others in [3,15,23,34,37]. Both the generalized Newton methods and the inexact Newton methods for solving nonsmooth equations are locally and superlinearly convergent to the solution under mild conditions.…”