Véra Lucia Rocha Lopes

Convergence properties of the inverse column-updating method

The inverse Column-Updating method is a secant algorithm for solving nonlinear systems of equations introduced recently by Martinez and Zambaldi (Optimization Methods and Software, 1 (1992), pp. 129-140). This method is one of the less expensive reliable quasi-Newton methods for solving nonlinear simultaneous equations, in terms of linear algebra work. Since it does not belong to the well-known LCSU (least-change secant-update) class, special arguments are used for proving local convergence. In this paper we prove that, if convergence is assumed, then R-superlinear convergence takes place. Moreover, we prove local convergence for a version of the method with (not necessarily Jacobian) restarts. Finally, we prove that local and R-superlinear convergence holds without restarts in the two-dimensional case. From a practical point of view, we show that, in some cases, the numerical performance of the inverse Column-Updating method is very good

Recent Applications and Numerical Implementation of Quasi-Newton Methods for Solving Nonlinear Systems of Equations

This paper presents a survey on recent applications of quasi-Newton methods to solve nonlinear systems of equations which appear in applied areas such as physics, biology, engineering, geophysics, chemistry and industry. It is also presented a comparative analysis of the performance of the ICUM (Inverse Column-Updating Method) and Broydens method when applied to some of the problems mentioned above.

Discrete Newton’s method with local variations for solving large-scale nonlinear systems

A globally convergent discrete Newton method is proposed for solving large-scale nonlinear systems of equations. Advantage is taken from discretization steps so that the residual norm can be reduced while the Jacobian is approximated, besides the reduction at Newtonian iterations. The Curtis–Powell–Reid (CPR) scheme for discretization is used for dealing with sparse Jacobians. Global convergence is proved and numerical experiments are presented.

On the local convergence of quasi-Newton methods for nonlinear complementarity problems

Abstract A family of Least-Change Secant-Update methods for solving nonlinear complementarity problems based on nonsmooth systems of equations is introduced. Local and superlinear convergence results for the algorithms are proved. Two different reformulations of the nonlinear complementarity problem as a nonsmooth system are compared, both from the theoretical and the practical point of view. A global algorithm for solving the nonlinear complementarity problem which uses the algorithms introduced here is also presented. Some numerical experiments show a good performance of this algorithm.

On the convergence of quasi-Newton methods for nonsmooth problems

We develop a theory of quasi-New ton and least-change update methods for solving systems of nonlinear equations F(x) = 0. In this theory, no differentiability conditions are necessary. Instead, we assume that Fcan be approximated, in a weak sense, by an affine function in a neighborhood of a solution. Using this assumption, we prove local and ideal convergence. Our theory can be applied to B-differentiable functions and to partially differentiable functions.

A globally convergent inexact Newton method with a new choice for the forcing term

Abstract In inexact Newton methods for solving nonlinear systems of equations, an approximation to the step sk of the Newton’s system J(xk)s=−F(xk) is found. This means that sk must satisfy a condition like ‖F(xk)+J(xk)sk‖≤ηk‖F(xk)‖ for a forcing term ηk∈[0,1). Possible choices for ηk have already been presented. In this work, a new choice for ηk is proposed. The method is globalized using a robust backtracking strategy proposed by Birgin et al. (Numerical Algorithms 32:249–260, 2003), and its convergence properties are proved. Several numerical experiments with boundary value problems are presented. The numerical performance of the proposed algorithm is analyzed by the performance profile tool proposed by Dolan and Moré (Mathematical Programming Series A 91:201–213, 2002). The results obtained show a competitive inexact Newton method for solving academic and applied problems in several areas.

A safeguard approach to detect stagnation of GMRES(m) with applications in Newton-Krylov methods

Restarting GMRES, a linear solver frequently used in numerical schemes, is known to suffer from stagnation. In this paper, a simple strategy is proposed to detect and avoid stagnation, without modifying the standard GMRES code. Numerical tests with the proposed modified GMRES(m) procedure for solving linear systems and also as part of an inexact Newton procedure, demonstrate the efficiency of this strategy.

Inverse q -columns updating methods for solving nonlinear systems of equations

In this work new quasi-Newton methods for solving large-scale nonlinear systems of equations are presented. In these methods q ( > 1) columns of the approximation of the inverse Jacobian matrix are updated in such a way that the q last secant equations are satisfied (whenever possible) at every iteration. An optimal maximum value for q that makes the method competitive is strongly suggested. The best implementation from the point of view of linear algebra and numerical stability is proposed and a local convergence result for the case q = 2 is proved. Several numerical comparative tests with other quasi-Newton methods are carried out.

Quasi-Newton acceleration for equality-constrained minimization

Abstract Optimality (or KKT) systems arise as primal-dual stationarity conditions for constrained optimization problems. Under suitable constraint qualifications, local minimizers satisfy KKT equations but, unfortunately, many other stationary points (including, perhaps, maximizers) may solve these nonlinear systems too. For this reason, nonlinear-programming solvers make strong use of the minimization structure and the naive use of nonlinear-system solvers in optimization may lead to spurious solutions. Nevertheless, in the basin of attraction of a minimizer, nonlinear-system solvers may be quite efficient. In this paper quasi-Newton methods for solving nonlinear systems are used as accelerators of nonlinear-programming (augmented Lagrangian) algorithms, with equality constraints. A periodically-restarted memoryless symmetric rank-one (SR1) correction method is introduced for that purpose. Convergence results are given and numerical experiments that confirm that the acceleration is effective are presented.

A comparative analysis of the monotone iteration method for elliptic problems

In this work, after a theoretical explanation of the monotone iteration method, there are presented several numerical experiments with this method, when applied to solve some nonlinear elliptic equations. It is shown that, in some cases, uniqueness of solution can also be verified through the numerical implementation of the method. It is also presented its application to cooperative elliptic systems. For all the examples Newtons method is also applied and a comparison between the monotone iteration method and Newtons method is made.