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 prese...

It is proposed that we use the term “approximation” for inexact description of a target system and “idealization” for another system whose properties also provide an inexact description of the target system. Since systems generated by a limiting process can often have quite unexpected, even inconsistent properties, familiar limit processes used in statistical physics can fail to provide idealiz...

An affine invariant convergence analysis for inexact augmented Lagrangian-SQP methods is presented. The theory is used for the construction of an accuracy matching between iteration errors and truncation errors, which arise from the inexact linear system solvers. The theoretical investigations are illustrated numerically by an optimal control problem for the Burgers equation.

The paper aims to extend the theory and application of nonconvex Newton-type methods, namely trust region cubic regularization, settings in which, addition solution subproblems, gradient Hessian objective function are approximated. Using certain conditions on such approximations, establishes optimal worst-case iteration complexities as exact counterparts. This is part a broader research program...

For the Hermitian inexact Rayleigh quotient iteration (RQI), we consider the local convergence of the inexact RQI with the Lanczos method for the linear systems involved. Some attractive properties are derived for the residual, whose norm is ξk, of the linear system obtained by the Lanczos method at outer iteration k + 1. Based on them, we make a refined analysis and establish new local converg...

In this note, we discuss the convergence behavior of a modified inexact Uzawa algorithm for solving generalized saddle point problems, which is an extension of the result obtained in a recent paper [Z.H. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math., 46 (2003) 157-171]. Keywords—Saddle point problem; Inexact Uzawa algorithm; Convergence behavior.

We present a general analytical model which describes the superlinear convergence of Krylov subspace methods. We take an invariant subspace approach, so that our results apply also to inexact methods, and to non-diagonalizable matrices. Thus, we provide a unified treatment of the superlinear convergence of GMRES, Conjugate Gradients, block versions of these, and inexact subspace methods. Numeri...

We propose in this paper an inexact dual gradient algorithm based on augmented Lagrangian theory and inexact information for the values of dual function and its gradient. We study the computational complexity certification of the proposed method and we provide estimates on primal and dual suboptimality and also on primal infeasibility. We also discuss implementation aspects of the proposed algo...