An Improvement of Global Error Bound for the Generalized Nonlinear Complementarity Problem over a Polyhedral Cone

Abstract

We consider the global error bound for the generalized nonlinear complementarity problem over a polyhedral cone (GNCP). By a new technique, we establish an easier computed global error bound for the GNCP under weaker conditions, which improves the result obtained by Sun and Wang (2013) for GNCP.

1. Introduction

Let be a polyhedral cone in for matrices , , and let be its dual cone; that is, For continuous mappings, , the generalized nonlinear complementarity problem, abbreviated as GNCP, is to find vector such that Throughout this paper, the solution set of the GNCP, denoted by , is assumed to be nonempty.

The GNCP is a direct generalization of the classical nonlinear complementarity problem and a special case of the general variational inequalities problem [1]. The GNCP was deeply discussed [2–7] after the work in [8]. It is well known that the global error bound is an important tool for theoretical analysis and numerical treatment for a mathematical problem [9, 10]. The global error bound estimation for GNCP with the mapping being -strongly -monotone was discussed in [4], a global error bound estimation for GNCP with the mapping being -strongly monotone and Hölder continuous was established in [5], and a global error bound for the GNCP for the linear and monotonic case was also established in [6, 7].

This paper is a follow-up to [4, 5, 11], as in these papers we will establish the global error bound estimation of the GNCP under weaker conditions than that needed in [4, 5, 11]. Based on a new technique, we establish a global error bound for the GNCP in terms of an easier computed residual function. The results obtained in this paper can be taken as an improvement of the existing results for GNCP and variational inequalities problem [4, 5, 11–13].

To end this section, we give some notations used in this paper. Vectors considered in this paper are taken in Euclidean space equipped with the usual inner product, and the Euclidean 2-norm and 1-norm of vector in are, respectively, denoted by and . We use to denote the nonnegative orthant in and use and to denote the vectors composed by elements , and , respectively. For simplicity, we use to denote vector , use to denote the identity matrix with appropriate dimension, use to denote a nonnegative vector , and use to denote the distance from point to the solution set .

2. Global Error Bound for the GNCP

First, we give the following definition used in the subsequent.

Definition 1. The mapping is said to be -uniform -function with respect to if there are constants and such that

Remark 2. Based on this definition, -uniform -function with respect to is weaker than -strongly -monotonicity by Definition 1 in [4], and if with , , then the above definition is equivalent in which the matrix is a -matrix [14]. For example, let By Theorem 2.1.15 in [14], it is easy to verify that is a -matrix. However, letting , we note that which shows that is not positive definite; that is, is not strongly monotonicity with respect to mapping .
Now, we give some assumptions for our analysis based on Definition 1.

Assumption 3. For mappings and matrix involved in the GNCP, we assume that(A1)mapping is -uniform -function with respect to mapping ;(A2)matrix has full-column rank.

In the following, we give the conclusion established in [2].

Theorem 4. A point is a solution of the GNCP if and only if there exist and , such that

From Theorem 4, under Assumption 3 (A2), similar to discussion in [4], we can transform the system (6) into the following system in which neither nor is involved: where

For the ease of description, let and . Then, system (7) can be written as where the solution set of (9) is denoted by .

In the following, we give the error bound for a single quadratic function to reach our aims.

Lemma 5. Let . Then, one has where is a constant, , and .

Proof. For any , let , , and Set , , , and . Obviously, this linear transformation is an invertible; that is, there exists an invertible matrix such that , and one has Without loss of generality, we assume . Define
It is easy to verify that and Let , and one has . Therefore, . Moreover, one has where the third equality follows from the definition of , the sixth and tenth equations are due to the definition of , respectively, the second inequality follows from the fact that and the third inequality follows from the fact that And, letting , then the desired result follows.

To establish a global error bound for GNCP, we also give the following result from [15] on the error bound for a polyhedral cone.

Lemma 6. For polyhedral cone with , , , and , there exists a constant such that

Before proceeding, we present the following definition introduced in [16] with constant .

Definition 7. The mapping is said to be -strongly nonexpanding with a constant , if , where .

Now, we are at the position to state our main results in this paper.

Theorem 8. Suppose that is -uniform -function with respect to mapping with positive constants and , respectively, and is -strongly nonexpanding with positive constants and , respectively. Then there exists constant such that

Proof. Using Lemma 5, for any , there exists such that where is defined in Lemma 5. Let From (9), we have . For convenience, we also let Using Lemma 6, for any , there exists such that where is a positive constant and the third and fourth inequalities follow from the fact that , for all .
Furthermore, where the second equality follows from the fact that and the first inequality is by nonexpanding property of projection operator. Combining (24), one has Combining (23) with (26), for any , we have where the second inequality follows from (23) with constants and , the third inequality uses (26), the fifth inequality follows from (20), the sixth inequality follows from the fact that the seventh and ninth inequalities follow from the fact that and the last inequality follows by letting .
For any , let . Then there exists such that where the second inequality follows from Definition 7 with constant , the third inequality follows from Definition 1 with constants and , the fifth inequality follows from the fact that , for all , and the sixth inequality is by (27). By (30) and letting , then the desired result follows.

Remark 9. Firstly, from remark of Definition 1, the conditions that is -uniform -function with respect to mapping and is -strongly nonexpanding in Theorem 8 are weaker than the conditions that is -strongly -monotone and is strongly nonexpanding (i.e., ) in Theorem 13 in [4]. In addition, the result in Theorem 8 is stronger than that in Theorem 13 in [4]. Thus, Theorem 8 is stronger than Theorem 13 in [4].
In the following, we also present an example to compare the condition in Theorem 8 in this paper and that in Theorem 13 in [4].

Example 10. When , , and , the (2) reduces to the linear complementarity problem (LCP) of finding vector such that where .

It is easy to see that the solution set of the LCP . In fact, Clearly, is a matrix [17]FV. Thus, there exists constant such that For any , . By Theorem 8, with , , , , , and and letting , we can obtain as . Thus, Theorem 8 provides a global error bound for this LCP.

However, letting , we note that which shows that is not positive definite, so the condition that is strongly monotone in Theorem 13 in [4] does not hold. Thus, the result of Theorem 13 in [4] fails in providing an error bound for this LCP.

Secondly, if is -strongly -monotone and is strongly -nonexpanding, then it is easy to verify that where is constant. Moreover, the conditions that both and are Hölder continuous (or both and are Lipschitz continuous) in Theorem 8 in this paper are removed. Thus, Theorem 8 is stronger than Theorem 2.5 in [5]. Furthermore, by Theorem 2.1 in [5], the GNCP can be reformulated as general variational inequalities problem, and the conditions in Theorem 8 are also weaker than those in Theorem 3.1 in [11], Theorem 2 in [12], Theorem 3.1 in [13], and Theorem 3.1 in [16], respectively.

In the end of this paper, we will consider a special case of GNCP which was discussed in [11].

When , the (2) reduces to the generalization of the classical nonlinear complementarity problem of finding vector such that Combining this with Theorem 8, we can also immediately obtain the following conclusion.

Corollary 11. Suppose that the hypotheses of Theorem 8 hold. Then there exists constant such that

Remark 12. It is clear that if is -uniform -function and is strongly -nonexpanding, for any , then so the condition in Corollary 11 is largely extended than the condition that is a uniform -function with respect to (i.e., ) in Theorem 3.3 in [11]. Moreover, the conditions that both and are Lipschitz continuous in Theorem 3.3 in [11] are removed. Thus, Corollary 11 is stronger than Theorem 3.3 in [11].

3. Conclusion

In this paper, we established a global error bound on the generalized nonlinear complementarity problems over a polyhedral cone, which improves the result obtained for variational inequalities and the GNCP in [4, 5, 11–13] by weakening the assumptions. Surely, we may use the error bound estimation to establish quick convergence rate of the methods for the GNCP under milder conditions. This is a topic for future research.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors wish to give their sincere thanks to the associated editor and two anonymous referees for their valuable suggestions and helpful comments which improve the presentation of the paper. This work was supported by the Natural Science Foundation of China (nos. 11171180, 11101303, 11171362, and 11271226), the Specialized Research Fund for the Doctoral Program of Chinese Higher Education (20113705110002, and 20120191110031), the Shandong Provincial Natural Science Foundation (ZR2010AL005), the Shandong Province Science and Technology Development Projects (2013GGA13034), the Domestic Visiting Scholar Project for the Outstanding Young Teacher of Shandong Province Universities (2013), and the Applied Mathematics Enhancement Program of Linyi University.