Journal of Applied Mathematics

Volume 2012 (2012), Article ID 245458, 11 pages

doi:10.1155/2012/245458

Research Article

Global Error Bound Estimation for the Generalized Nonlinear Complementarity Problem over a Closed Convex Cone

Hongchun Sun¹ and Yiju Wang²

¹School of Sciences, Linyi University, Shandong, 276005 Linyi, China

²School of Management Science, Qufu Normal University, Shandong, 276800 Rizhao, China

Received 10 February 2012; Accepted 28 April 2012

Academic Editor: Zhenyu Huang

Copyright © 2012 Hongchun Sun and Yiju Wang. 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.

The global error bound estimation for the generalized nonlinear complementarity problem over a closed convex cone (GNCP) is considered. To obtain a global error bound for the GNCP, we first develop an equivalent reformulation of the problem. Based on this, a global error bound for the GNCP is established. The results obtained in this paper can be taken as an extension of previously known results.

1. Introduction

Let mappings , , and the generalized nonlinear complementarity problem, abbreviated as GNCP, is to find vector such that

where is a nonempty closed convex cone in and is its dual cone, that is, . We denote the solution set of the GNCP by , and assume that it is nonempty throughout this paper.

The GNCP is a direct generalization of the classical nonlinear complementarity problem which finds applications in engineering, economics, finance, and robust optimization operations research [1–3]. For example, the balance of supply and demand is central to all economic systems; mathematically, this fundamental equation in economics is often described by a complementarity relation between two sets of decision variables. Furthermore, the classical Walrasian law of competitive equilibria of exchange economies can be formulated as a generalized nonlinear complementarity problem in the price and excess demand variables [2]. Up to now, the issues of numerical methods and existence of the solution for the problem were discussed in the literature [4].

Among all the useful tools for theoretical and numerical treatment to variational inequalities, nonlinear complementarity problems, and other related optimization problems, the global error bound, that is, an upper bound estimation of the distance from a given point in to the solution set of the problem in terms of some residual functions, is an important one [5, 6]. The error bound estimation for the generalized linear complementarity problems over a polyhedral cone was analyzed by Sun et al. [7]. Using the natural residual function, Pang [8] obtained a global error bound for the strongly monotone and Lipschitz continuous classical nonlinear complementarity problem with a linear constraint set. Xiu and Zhang [9] also presented a global error bound for general variational inequalities with the mapping being strongly monotone and Lipschitz continuous in terms of the natural residual function. If , is -strongly monotone and Hölder continuous, the local error bound for classical variational inequality problems was given by Solodov [6].

To our knowledge, the global error bound for the problem (1.1) with the mapping being -strongly monotone and Hölder-continuous hasn't been investigated. Motivated by this fact, The main contribution of this paper is to establish a global error bound for the GNCP via the natural residual function under milder conditions than those needed in [6, 8, 9]. The results obtained in this paper can be taken as an extension of the previously known results in [6, 8, 9].

We give some notations used in this paper. Vectors considered in this paper are all taken in Euclidean space equipped with the standard inner product. The Euclidean norm of vector in the space is denoted by . The inner product of vector in the space is denoted by .

2. The Global Error Bound for GNCP

In this section, we would give error bound for GNCP, which can be viewed as extensions of previously known results. To this end, we will in the following establish an equivalent reformulation of the GNCP and state some well-known properties of the projection operator which is crucial to our results.

In the following, we first give the equivalent reformulation of the GNCP.

Theorem 2.1. A point is a solution of (1.1) if and only if is a solution of the following problem:

Proof. Suppose that is a solution of (2.1). Since vector , by substituting into (2.1), we have . On the other hand, since , then . By substituting into (2.1), we obtain . Consequently, . For any , we have , that is, . Combining , thus, is a solution of (1.1).

On the contrary, suppose that is a solution of (1.1), since , for any , we have , and from , we have , combining . Therefor, is a solution of (2.1).

Now, we give the definition of projection operator and some related properties [10]. For nonempty closed convex set and any vector , the orthogonal projection of onto , that is, , is denoted by .

Lemma 2.2. For any , then

(i), (ii).

For (2.1), is a constant, is called projection-type residual function, and let . The following conclusion provides the relationship between the solution set of (2.1) and that of projection-type residual function [11], which is due to Noor [11].

Lemma 2.3. is a solution of (2.1) if and only if .

To establish the global error bound of GNCP, we also need the following definition.

Definition 2.4. The mapping is said to be

(1)-strongly monotone with respect to if there are constants such that

(2)Hölder-continuous if there are constants such that

In this following, based on Lemmas 2.2 and 2.3, we establish error bound for GNCP in the set .

Theorem 2.5. Suppose that is -strongly monotone with respect to and with positive constants , both and are continuous with positive constants , respectively, and holds. Then for any , there exists a solution of (1.1) such that

Proof. Since

by the first inequality of (2.1),

Combining with Lemma 2.2(i), we have

Substituting in (2.9) by leads to that

Using (2.8) and (2.10), we obtain

that is,

Base on Definition 2.4, a direct computation yields that

Combining this, we have

On the other hand, for the first inequality of (2.1), by Lemmas 2.3 and 2.2(ii), we have

Thus,

Combining (2.14) with (2.16) for any , if , then

For any , if , then

If , then , by (2.17), we have , and using (2.17) again, we obtain that (2.4) holds.

If , then , combining this with (2.18), we have , and using (2.18) again, we conclude that (2.5) holds.

If , then

Combining (2.17) with (2.18), we conclude that (2.6) holds.

Definition 2.6. The mapping involved in the GNCP is said to be -strongly monotone in if there are positive constants such that

Base on Theorem 2.5, we are at the position to state our main results in the following.

Theorem 2.7. Suppose that the hypotheses of Theorem 2.5 hold, is -strongly monotone, and the set is convex. Then, there exists a constant , such that, for any , there exists such that

where

Proof. For given , we only need to first project to , that is, there exists a vector such that . By Definition 2.6, there exist constants such that

that is, .

Since

Combining this, we have

Combining (2.26) with Theorem 2.5, we have the following results.

Case 1 (if and ). Combining (2.4) with the first inequality in (2.26), we can obtain that

where ,.

Case 2 (If and ). Combining (2.4) with the second inequality in (2.26), we can also obtain that

where ,.

Case 3 (if and ). Combining (2.5)-(2.6) with the first inequality in (2.26), we can obtain that

where ,.

Case 4 (If and ). Combining (2.5)-(2.6) with the second inequality in (2.26), we can also obtain that

where ,.

By (2.27)–(2.30), we can deduce that (2.21) holds.

Based on Theorem 2.7, we can further establish a global error bound for the GNCP. First, we give that the needed result from [12] mainly discusses the error bound for a polyhedral cone to reach our claims.

Lemma 2.8. For polyhedral cone with , there exists a constant such that

Theorem 2.9. Suppose that the hypotheses of Theorem 2.5 hold, and is linear mapping. Then, there exists a constant , such that, for any , there exists such that

where

Proof. For given , we only need to first project to , that is, there exists a vector such that . By Lemma 2.8, there exists a constant such that . In the following, the proof is similar to that of Theorem 2.7, and we can deduce that (2.32) holds.

Remark 2.10. If the constraint condition is removed in (1.1), is strongly monotone with respect to (i.e., ), and both and are Lipschitz continuous (i.e., ), the error bound in Theorems 2.5, 2.7, and 2.9 reduces to result of Theorem 3.1 in [9].

If the constraint condition is removed in (1.1) and , is strongly monotone (i.e., ) and Lipschitz continuous (i.e., ), the error bound in Theorems 2.5, 2.7, and 2.9 reduces to result of Theorem 3.1 in [8].

If the constraint condition is removed in (1.1) and , is -strongly monotone in set and Hölder continuous, the error bound in Theorems 2.5, 2.7, and 2.9 reduces to result of Theorem 2 in [6].

Acknowledgments

The authors wish to give their sincere thanks to the anonymous referees for their valuable suggestions and helpful comments which improved the presentation of the paper. This work was supported by the Natural Science Foundation of China (Grant nos. 11171180, 11101303), Specialized Research Fund for the Doctoral Program of Chinese Higher Education (20113705110002), and Shandong Provincial Natural Science Foundation (ZR2010AL005, ZR2011FL017).

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