Abstract and Applied Analysis

Volume 2014 (2014), Article ID 745981, 5 pages

http://dx.doi.org/10.1155/2014/745981

Research Article

Generalized Newton Method for a Kind of Complementarity Problem

Shou-qiang Du

College of Mathematics, Qingdao University, Qingdao 266071, China

Received 20 February 2014; Accepted 6 April 2014; Published 27 April 2014

Academic Editor: Yisheng Song

Copyright © 2014 Shou-qiang Du. 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.

A generalized Newton method for the solution of a kind of complementarity problem is given. The method is based on a nonsmooth equations reformulation of the problem by F-B function and on a generalized Newton method. The merit function used is a differentiable function. The global convergence and superlinear local convergence results are also given under suitable assumptions. Finally, some numerical results and discussions are presented.

1. Introduction

This paper considers a kind of complementarity problem:

where are two differentiable functions. Equation (1) is also denoted by GCP(F, G) in [1]. When , (1) reduces to the nonlinear complementarity problem, which is a general framework for optimality conditions of nonlinear optimization problems as well as some problems arising in different fields. In the past few years, several theoretical and computational results for complementarity have been established, such as [1–12]. The method is based on semismooth equations reformulation of (1). As we know, Newton-type methods are one of the fastest methods for the solution of equations and nonsmooth equations. But the methods are usually only locally convergent. In recent years, some research work has been devoted to techniques for globalizing the local methods. So, in this paper, we are interested in solving (1) by a Newton based method. The method is based on a line search and a semismooth equations reformulation of (1).

This paper is organized as follows. In the next section, (1) is converted into a semismooth equation. Then, a generalized Newton method for solving the reformulation is introduced. The global convergence and the local superlinear convergence of the method are also presented. In the Section 3, numerical experimental results and some discussions are listed.

2. Preliminaries and Method

Firstly, we introduce some notations and propositions used in the paper (see for instance [1–5]). Given , we use to denote its Jacobian at . If is locally Lipschitzian and denote the set of differentiable points, the B-subdifferential and Clarke general Jacobian of at are defined as

Let be a locally Lipschitz function. If the following limit,

exists for any , is said to be semismooth at .

Proposition 1. If is semismooth at , the following equations are equivalent.

(a)When ,

(b)Where ,

Definition 2. is called a -matrix if its principal minors are all positive and a -matrix if its principal minors are all nonnegative.

Definition 3. is called function if is continuously differentiable and its gradient is semismooth.

Definition 4. is BD-regular at if are nonsingular.

In the following, we give the reformulation of (1) and the generalized Newton method. Based on the F-B function

we consider the following reformulation of (1):

solves (1) solves (7). Denoting

we know that is locally Lipschitz on and differentiable on . When , we have

where . We also know that

When is not in , for , we have

where .

Then, for , we have

Defining a merit function as

then solving (1) is equivalent to solving the following unconstrained minimization problem:

where is continuously differentiable. We know that , where . Now, we give the following generalized Newton method for solving (1).

Method 1 (generalized Newton method). Given and , .

Step 1. If , stop.

Step 2. Choose and compute of the equation

If the condition

is not satisfied, set .

Step 3. Find the smallest integer such that

Set , let , and go to Step 1.

The direction obtained by solving (15) is a descent direction for the function . This is a standard property of Newton method for the solution of a smooth equation. But it is no longer true for nonsmooth equations. In the above method, the direction obtained by solving (15) is a descent direction by the fact that . On the other hand, as we know, is not unique in (15) if is not nonsingular. So, we will give the following proposition to ensure is nonsingular.

Proposition 5. Suppose that given by (1) satisfy , for any and ; then is nonsingular.

Proof. Let and let solve . By the above analysis of , for constants , we have

and so we have

Multiplying the th equation in the above equations by , we get

If , by (12), we have . So,

and then

We have

Then . On the other hand, if , we have

By , we get

Then and the nonsingularity of follows. This completes the proof.

may have a global minimum with when (1) has no solution. Another point which is worth concern is that the conditions of a stationary point of are a solution of (1). Similar to [1], we give the conditions, which guarantee that every stationary point of is a solution of (1).

Proposition 6. If (1) has a nonempty solution set, then is a solution of (1) .

Proposition 7. If is a stationary point of   such that is nonsingular and is a matrix, then is a solution of (1).

Now, we give the global convergence theorem for Method 1.

Theorem 8. Suppose that the sequence is generated by Method 1. Then each accumulation point of the sequence is a stationary point of  .

Proof. The following proof is given by two parts.

Part I. If, for an infinite set of indices , we have , then, by Proposition 1.16 in [7], we know that any limit point of is a stationary point of .

Part II. We assume that the direction is always computed by (15). Suppose that and . By (15), we have

From (26), we get

For and , we know that

If for in . For which is bounded and (27), we get , which is contradicting the assumption. On the other hand, taking into account that is bounded and (16), cannot be unbounded. Because of (17) and the fact that is a continuously differentiable function, we have

In the following, we will prove that is bounded away from . Supposing the contrary,

By (28), we can assume that . Taking limit of both sides of (30), we have

From (16), we know , which contradicts (31). So, we know that is bounded away from . On the other hand, (16) and (29) imply that and this contradicts (28). This completes the proof.

In the following part of this section, we will discuss the local superlinear convergence of Method 1.

Remark 9. Suppose that the sequence is generated by Method 1. If one of the limit points of the sequence is , which is a solution of (1), and is a BD-regular solution of , then, we can prove locally that the direction is always the solution of (15) and eventually the step size of one satisfies (17). So the method eventually reduces to the local method .

Theorem 10. Let the BD-regular condition hold at an accumulation point of the sequence generated by Method 1. Then converges to Q-superlinearly. The rate of convergence is Q-quadratic if and are functions.

Proof. Since is an accumulation point of , there exists a subsequence such that . We know for all sufficiently large. Therefore, the results of this theorem are guaranteed by Theorem 3.1 in [8].

Remark 11. In Method 1, (17) can be replaced by the following line search:

where and , and is a integer.

3. Numerical Results and Discussions

In this section, we present some numerical results and also give some discussions about a method for calculating a generalized Jacobian of . For solving the systems of (7), we can take as a tool instead of the Clarke generalized Jacobian and B-differential. We give the following for in (7):

where , if , or if , and , if .

Proposition 12. Suppose that and are defined by (7) and by (33); then all are nonsingular.

Example 13. We consider the generalized complementarity problem (1), where the functions

Both and are continuously differentiable functions.

We use Method 1 to compute Example 13. Results for Example 13 with initial points and are presented in Table 1.

Table 1

Step

16

22

48

32

37

In Method 1, if we replace (17) by (32), we also can use this method to compute Example 13. Results for Example 13 with initial points and are presented in Table 2.

Table 2

Step

16

24

20

30

37

Discussion 1. From the numerical results for Method 1 in Tables 1 and 2, we can see that (17) work as well as (32). So we can use (17) or (32) in practice.

In Method 1, we also can replace (15) by (35) or (36). Then Method 1 becomes the Levenberg-Marquardt method and the modified Levenberg-Marquardt method, which were given in [4, 10]. was computed by

or

We use Method 1 ( was computed by (15)), the Levenberg-Marquardt method ( was computed by (35)), and the modified Levenberg-Marquardt method ( was computed by (36)) to compute Example 13. Results for Example 13 with initial points and are presented in Table 3.

Table 3

Step ( was computed by (12))

Step ( was computed by (35))

Step ( was computed by (36))

Step ( was computed by (12))

Step ( was computed by (35))

Step ( was computed by (36))

Discussion 2. From the numerical results for the Method 1 ( was computed by (15)), the Levenberg-Marquardt method ( was computed by (35)), and the modified Levenberg-Marquardt method ( was computed by (36)) in Table 3, we can see that our method in this paper works quite better than the methods in [4, 10].

From Discussion 1, we can see that (17) work as well as (32). When we use line search (32) to replace (17), we give the following numerical results for Method 1 ( was computed by (15)), the Levenberg-Marquardt method ( was computed by (35)), and the modified Levenberg-Marquardt method ( was computed by (36)) to compute Example 13. The numerical results are given in Table 4.

Table 4

Step ( was computed by (12))

Step ( was computed by (35))

Step ( was computed by (36))

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by National Science Foundation of China (11101231).

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