A smoothing inexact Newton method for nonlinear complementarity problems

In this article, we propose a new smoothing inexact Newton algorithm for solving nonlinear complementarity problems (NCP) base on the smoothed Fischer-Burmeister function. In each iteration, the corresponding linear system is solved only approximately. The global convergence and local superlinear convergence are established without strict complementarity assumption at the NCP solution. Preliminary numerical results indicate that the method is effective for large-scale NCP.     

A Newton inexact interior-point method for large scale nonlinear optimization problems

In this paper, we describe a variant of the Newton Interior-Point method in [8] for nonlinear programming problems. In this scheme, the perturbation parameter can be chosen within a range of, values and we can use an iterative method for approximately solving the reduced linear system arising at each step. We have devised the inner termination rule which guarantees the global convergence of this Newton Inexact Interior-Point method. We remark that the required assumptions are weaker than those stated in [8], as shown by some numerical examples. This research was supported by the Italian Ministry for Education, University and Research (MIUR), FIRB Project No. RBAU01JYPN.     

A nonsmooth Newton method for variational inequalities,I: Theory

This paper presents a modified damped Newton algorithm for solving variational inequality problems based on formulating this problem as a system of equations using the Minty map. The proposed modified damped-Newton method insures convergence and locally quadratic convergence under the assumption of regularity. Under the assumption ofweak regularity and some mild conditions, the modified algorithm is shown to always create a descent direction and converge to the solution. Hence, this new algorithm is often suitable for many applications where regularity does not hold. Part II of this paper presents the results of extensive computational testing of this new method.Corresponding author.     

A nonsmooth Newton method for variational inequalities,II: Numerical results

This paper presents the results of extensive computational testing of the modified damped Newton algorithm for solving variational inequality problems presented in Part I [8].Corresponding author.     

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

In inexact Newton methods for solving nonlinear systems of equations, an approximation to the step s _k of the Newton’s system J(x _k )s=−F(x _k ) is found. This means that s _k must satisfy a condition like ‖F(x _k )+J(x _k )s _k ‖≤η _k ‖F(x _k )‖ for a forcing term η _k ∈[0,1). Possible choices for η _k have already been presented. In this work, a new choice for η _k 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. Supported by FAPESP, CNPq, PRONEX-Optimization.     

An inexact Newton method for nonlinear two-point boundary-value problems

The method of quasilinearization for nonlinear two-point boundary-value problems is an application of Newton's method to a nonlinear differential operator equation. Since the linear boundary-value problem to be solved at each iteration must be discretized, it is natural to consider quasilinearization in the framework of an inexact Newton method. More importantly, each linear problem is only a local model of the nonlinear problem, and so it is inefficient to try to solve the linear problems to full accuracy. Conditions on size of the relative residual of the linear differential equation can then be specified to guarantee rapid local convergence to the solution of the nonlinear continuous problem. If initial-value techniques are used to solve the linear boundary-value problems, then an integration step selection scheme is proposed so that the residual criteria are satisfied by the approximate solutions. Numerical results are presented that demonstrate substantial computational savings by this type of economizing on the intermediate problems.This work was supported in part by DOE Contract DE-AS05-82-ER13016 and NSF Grant RII-89-17691 and was part of the author's doctoral thesis at Rice University. It is a pleasure to thank the author's thesis advisors, Professor R. A. Tapia and Professor J. E. Dennis, Jr.     

A new logarithmic-quadratic proximal method for nonlinear complementarity problems

In this paper, we propose a new modified logarithmic-quadratic proximal (LQP) method for solving nonlinear complementarity problems (NCP). We suggest using a prediction-correction method to solve NCP. The predictor is obtained via solving the LQP system approximately under significantly relaxed accuracy criterion and the new iterate is computed by using a new step size α_k. Under suitable conditions, we prove that the new method is globally convergent. We report preliminary computational results to illustrate the efficiency of the proposed method. This new method can be considered as a significant refinement of the previously known methods for solving nonlinear complementarity problems.     

Convergence of a generalized Newton and an inexact generalized Newton algorithms for solving nonlinear equations with nondifferentiable terms

In this paper, we consider two versions of the Newton-type method for solving a nonlinear equations with nondifferentiable terms, which uses as iteration matrices, any matrix from B-differential of semismooth terms. Local and global convergence theorems for the generalized Newton and inexact generalized Newton method are proved. Linear convergence of the algorithms is obtained under very mild assumptions. The superlinear convergence holds under some conditions imposed on both terms of equation. Some numerical results indicate that both algorithms works quite well in practice.      

A wavelet adaptive Newton method for the solution of nonlinear equations

For the solution of nonlinear equations, we present an adaptive wavelet scheme, which couples an inexact Newton method and the idea of nonlinear wavelet approximation. In particular, we obtain a result of quadratic convergence.     

A new semi-local convergence theorem for the inexact Newton methods

We present a new semi-local convergence theorem for the inexact Newton methods in the assumption that the derivative satisfies some kind of weak Lipschitz conditions. As special cases of our main result we re-obtain some well-known convergence theorems for Newton methods.