A smoothing and regularization predictor-corrector method for nonlinear inequalities

Che

Description

For a system of nonlinear inequalities, we approximate it by a family of parameterized smooth equations via a new smoothing function. We present a new smoothing and regularization predictor-corrector algorithm. The global and local superlinear convergence of the algorithm is established. In addition, the smoothing parameter μ and the regularization parameter ε in our algorithm are viewed as different independent variables. Preliminary numerical results show the efficiency of the algorithm. MSC: 90C33, 90C30, 15A06.

Sujets

• Smooth function

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 12
• Langue: English

Extrait

CheJournal of Inequalities and Applications2012,2012:214 http://www.journalofinequalitiesandapplications.com/content/2012/1/214

R E S E A R C HOpen Access A smoothing and regularization predictor-corrector method for nonlinear inequalities * Haitao Che

* Correspondence: [e-mail] School of Mathematics and Information Science, Weifang University, Weifang, Shandong 261061, China

Abstract For a system of nonlinear inequalities, we approximate it by a family of parameterized smooth equations via a new smoothing function. We present a new smoothing and regularization predictor-corrector algorithm. The global and local superlinear convergence of the algorithm is established. In addition, the smoothing parameter µ and the regularization parameter ε in our algorithm are viewed as different independent variables. Preliminary numerical results show the efficiency of the algorithm. MSC:90C33; 90C30; 15A06 Keywords:nonlinear inequalities; predictor-corrector algorithm;P 0 -function; smoothing function

1 Introduction Consider the following system of nonlinear inequalities:

f(x)≤,

(.)

n wheref(x) = (f  (x),f  (x), . . . ,f n (x)) andf i :R→Ris a continuously differentiable function fori= , , . . . ,n. This problem finds applications in data analysis, set separation problems, computer-aided design problems and image reconstructions [–]. Among various solu- tion methods for the inequality problems [–], the smoothing-type methods receive much attention [–] which first transform the problem as a system of nonsmooth equa- tions and approximate it by a smooth equation and then solve it by the smoothing Newton methods. Since the derivative of the underlying mapping may be seriously ill-conditioned, which may prevent the smoothing methods from converging to a solution of the prob- lem, a perturbed regularization technique is introduced to overcome this drawback [, , ]. In , Huanget al.proposed a predictor-corrector smoothing Newton method for nonlinear complementarity problem with aP  function based on the perturbed minimum function []. The method was shown to be locally superlinear convergent under the as- * sumptions that allV∈∂H(z) are nonsingular andf(x) is locally Lipschitz continuous * aroundx. In this paper, motivated by the smoothed penalty function for constrained optimiza- tion [], we construct a new smoothing function for nonlinear inequalities, and thus we

©2012 Che; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.