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Mathematical Problems in Engineering
Volume 2008, Article ID 942391, 10 pages
http://dx.doi.org/10.1155/2008/942391
Research Article

A Modified Levenberg-Marquardt Method for Nonsmooth Equations with Finitely Many Maximum Functions

1School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China
2College of Mathematics, Qingdao University, Qingdao 266071, China

Received 2 August 2008; Accepted 26 November 2008

Academic Editor: Shijun Liao

Copyright © 2008 Shou-qiang Du and Yan Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For solving nonsmooth systems of equations, the Levenberg-Marquardt method and its variants are of particular importance because of their locally fast convergent rates. Finitely many maximum functions systems are very useful in the study of nonlinear complementarity problems, variational inequality problems, Karush-Kuhn-Tucker systems of nonlinear programming problems, and many problems in mechanics and engineering. In this paper, we present a modified Levenberg-Marquardt method for nonsmooth equations with finitely many maximum functions. Under mild assumptions, the present method is shown to be convergent Q-linearly. Some numerical results comparing the proposed method with classical reformulations indicate that the modified Levenberg-Marquardt algorithm works quite well in practice.

1. Introduction

In the past few years, there has been a growing interest in the study of nonlinear equations (see, e.g., [1, 2]) and nonsmooth equations, which have been proposed in the study of the nonlinear complementarity problem, the variational inequality problem, equilibrium problem and engineering mechanics (see, e.g., [310]).

Finitely many maximum functions systems are very useful in the study of nonlinear complementarity problems, variational inequality problems, Karush-Kuhn-Tucker systems of nonlinear programming problems, and many problems in mechanics and engineering. In the present paper, we study a new method for nonsmooth equations with finitely many maximum functions system proposed in [11] where for are continuously differentiable, for are finite index sets. Denote where Then (1.1) can be rewritten as follows:where is a nonsmooth function. By using the following subdifferential for the function given in (1.2),Gao gave Newton method for (1.4) with the superlinear convergence in [11].

Based on [5, 11], we present a modification of the Levenberg-Marquardt method for solving nonsmooth equations. In Section 2, we recall some results of generalized Jacobian and semismoothness. In Section 3, we give the Levenberg-Marquardt method which has been proposed in [5] and the new modified Levenberg-Marquardt method for the system of nonsmooth equations with finitely many maximum functions. The convergence of the modified Levenberg-Marquardt algorithm is also given. In Section 4, some numerical tests comparing the proposed modified Levenberg-Marquardt algorithm with the original method show that our algorithm works quite well.

2. Preliminaries

We start with some notions and propositions, which can be found in [811].

Let be locally Lipschitzian. Then, is almost everywhere F-differentiable. Let the set of points where is F-differentiable be denoted by . Then for ,The general Jacobian of at in the sense of Clarke is defined as

Proposition 2.1. is a nonempty and compact set for any the point to set -subdifferential map is upper semicontinuous.

Proposition 2.2. is a nonempty and compact set for any and upper semicontinuous.

Proof. From the fact that is a finite set of points in and can be calculated by determining the index sets and evaluating the gradients ,

Definition 2.3. is semismooth at if is locally Lipschitz at andexists for all . If is semismooth at , one knows . If for all , , one calls the function is strongly semismooth at .

Proposition 2.4. (I) If is locally Lipschitz continuous and semismooth at , then
(II) If is locally Lipschitz continuous, strongly semismooth at , and directionally differentiable in a neighborhood of , then

Lemma 2.5. Equation of maximum functions (1.4) is a system of semismooth equations.

In the study of algorithms for the local solution of semismooth systems of equations, similar to [11], one also has the following lemmas.

Lemma 2.6. Suppose that and are defined by (1.4) and by (1.5), respectively, and all are nonsingular. Then there exists a constant such that

The proof is similar to [11, Lemma 2.1], from the fact that is a finite set of points.

Lemma 2.7. Suppose that is a solution of (1.1), thenfor all in some neighborhood of and and for ,

Since each of (1.1) is continuous, one gets the lemma immediately.

3. Modified Levenberg-Marquardt Method and Its Convergence

In this section, we briefly recall some results on the Levenberg-Marquardt-type method for the solution of nonsmooth equations and their local convergence (see, e.g., [5, 9]). We also give the modified Levenberg-Marquardt method and analyze its local behavior. Now we consider exact and inexact versions of Levenberg-Marquardt method.

Given a starting vector , letwhere is the solution of the systemIn the inexact versions of this method can be given by the solution of the systemwhere is the vector of residuals and we can assume for some .

We now give the modified Levenberg-Marquardt method for (1.1) as follows.

Modified Levenberg-Marquardt Method
Step 1. Given , .
Step 2. Solve the system to get ,for and is the vector of residuals
Step 3. Set , if , terminate. Otherwise, let , and go to Step 2.
Based upon the above analysis, we give the following local convergence result.

Theorem 3.1. Suppose that is a sequence generated by the above method and there exist constants , for all . Let be a solution of , and let all be nonsingular. Then the sequence converges Q-linearly to for .
Proof. By Lemma 2.6 and the continuously differentiable of , there is a constant such that for all sufficiently close to are nonsingular withFurthermore, by Proposition 2.4, there exists , which can be taken arbitrarily small, such thatfor all in a sufficiently small neighborhood of depending on . By Proposition 2.2 the upper semicontinuity of the , we also knowfor all and all sufficiently close to , with being a suitable constant. From the locally Lipschitz continuous of , we havefor all in a sufficiently small neighborhood of and a constant . From (3.4), we also knowMultiply the above equation by and taken into account Lemma 2.7, and (3.6), (3.7), (3.8), and (3.9), we getLet , soSince can be chosen arbitrarily small, by taking sufficiently close to , there exist and such that , so that the Q-linear convergence of to follows by taking for a small enough . Thus we complete the proof of the theorem.

Theorem 3.2. Suppose that is a sequence generated by the above method and there exist constants , for all . Then the sequence converges Q-linearly to for .

The proof is similar to that of Theorem 3.1, so we omit it.

Following the proof of Theorem 3.1, the following statement holds.

Remark 3.3. Theorems 3.1 and 3.2 hold with in (3.4).

4. Numerical Test

In order to show the performance of the modified Levenberg-Marquardt method, in this section, we present numerical results and compare the Levenberg-Marquardt method and modified Levenberg-Marquardt method. The results indicate that the modified Levenberg-Marquardt algorithm works quite well in practice. All the experiments were implemented in Matlab 7.0.

Example 4.1. whereFrom (1.1), we knowwhere .

Our subroutine computes such that (3.3) and (3.4) hold with . We also use the condition as the stopping criterion. We can see that our method is good for Example 4.1.

Results for Example 4.1 with initial point and computes by (3.4) are listed in Table 1.

tab1
Table 1: and computes by (3.4).

Results for Example 4.1 with initial point and computes by (3.3) are listed in Tables 2 and 3.

tab2
Table 2: and computes by (3.3).
tab3
Table 3: and computes by (3.3).

Results for Example 4.1 with initial point and computes by (3.4) are listed in Tables 4 and 5.

tab4
Table 4: and computes by (3.4).
tab5
Table 5: and computes by (3.4).

Results for Example 4.1 with initial point and computes by (3.3) are listed in Tables 6 and 7.

tab6
Table 6: and computes by (3.3).
tab7
Table 7: and computes by (3.3).

Results are shown for Example 4.1 with initial point . We also use the condition as the stopping criterion and computes by (3.4) we get that by 21 steps . When we compute by (3.3), we get that by 45 steps . We can test the method with other examples and will think the global convergence of the method in another paper.

Acknowledgments

This work was supported by National Science Foundation of China (under Grant: 10671126). The Innovation Fund Project for Graduate Student of Shanghai (JWCXSL0801) and key project for Fundamental Research of STCSM (Project no. 06JC14057) and Shanghai Leading Academic Discipline Project (S30501). The authors are also very grateful to referees for valuable suggestions and comments.

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