A smoothing and regularization predictor-corrector method for nonlinear inequalities

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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.

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Smooth function

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 Publié par Publié le 01 janvier 2012 Nombre de lectures 12 Langue English
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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: haitaoche@163.com 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 parameterµ and the regularization parameterεin 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;P0-function; smoothing function
1 Introduction Consider the following system of nonlinear inequalities:
f(x),
(.)
n wheref(x) = (f(x),f(x), . . . ,fn(x)) andfi:RRis 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 aPfunction based on the perturbed minimum function []. The method was shown to be locally superlinear convergent under the as-*sumptions that allVH(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.
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