Abstract and Applied Analysis

Volume 2013 (2013), Article ID 209340, 13 pages

http://dx.doi.org/10.1155/2013/209340

Research Article

A Sharper Global Error Bound for the Generalized Nonlinear Complementarity Problem over a Polyhedral Cone

Hongchun Sun¹ and Yiju Wang²

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

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

Received 12 July 2013; Revised 17 September 2013; Accepted 30 September 2013

Academic Editor: Abdellah Bnouhachem

Copyright © 2013 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.

We revisit the global error bound for the generalized nonlinear complementarity problem over a polyhedral cone (GNCP). By establishing a new equivalent formulation of the GNCP, we establish a sharper global error bound for the GNCP under weaker conditions, which improves the existing error bound estimation for the problem.

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 [2–5] after the work in [6]. The GNCP plays a significant role in economics, operation research, nonlinear analysis, and so forth (see [7, 8]). For example, 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 (see [8]).

For the GNCP, the solution existence and the numerical solution methods for the GNCP were discussed [2, 3, 6]. As an important tool for a mathematical problem, the global error bound estimation for GNCP with the mapping being -strongly monotone and Hölder continuous was discussed in [5], and a global error bound for the GNCP for the linear and monotonic case was established in [4].

In this paper, we will establish a global error bound for the problem (2) without the Hölder continuity of the underlying mapping. To this end, we first develop some new equivalent reformulations of the GNCP under weaker conditions and then establish a sharper global error bound for the GNCP in terms of some easier computed residual functions. The results obtained in this paper can be taken as an improvement of the existing results for GNCP and variational inequalities problem [4, 5, 9–11].

To end this section, we give some notations used in this paper. Vectors considered in this paper are taken in the 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 , , , 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 some concepts used in the subsequent.

Definition 1. The mapping is said to be

(i)monotone with respect to if

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

Remark 2. Based on this definition, -strongly -monotone implies monotonicity, and if , with , , then the above Definition 1(i) is equivalent to that the matrix is positive semidefinite.

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 monotone with respect to mapping ;(A2) matrix has full-column rank.

Remark 4. Under (A2) in the assumption, matrix has left inverse , that is, its pseudoinverse of . Certainly, the assumption on matrix is weaker than that on matrix which has full-column rank [4]. In addition, when the mappings , are both linear, then Assumption 3(A1) coincides with Assumption (A1) in [4].

In the following, we will establish a new equivalent reformulation to the GNCP. First, we give the following conclusion established in [2].

Theorem 5. A point is a solution of the GNCP if and only if there exist , , such that

From Theorem 5, under Assumption 3(A2), we can transform the system into a new system in which neither nor is involved. To this end, we need the following conclusion [12].

Lemma 6. If the linear system is consistent, then is the solution with the minimum 2-norm, where is the pesudo-inverse of .

Lemma 7. Suppose that Assumption 3(A2) holds. Then, for any , the following statements are equivalent.

(1) There exist , such that .(2) Consider

where .

Proof. The proof follows that of Lemma 2.1 in [4], and for completeness, we include it.

Set

Now, we show that these two sets are equal.

First, for any , there exist , such that

Premultiplying (8) by gives

Combining this with (8) yields that

that is,

Recalling Lemma 6, we further have

Combining this with (9) yields that

Using (8), (12), and (13), we have

From the fact that , by (13), one has

Combining this with (14) leads to that . This shows that .

Second, for any , let

Then, , . From (14), one has

that is, . Hence, , and the desired result follows.

Combining this conclusion with Theorem 5, we can establish the following equivalent formulation of the GNCP:

where

For the ease of description, we denote , . Thus, system (18) can be written as

For system (20), one has

where the first equality follows from the last equality in (20), and the last equality uses the second equality in (20). Thus, system (20) can be further written as

Furthermore, for any with , , it holds from (21) that

Now, consider the following optimization problem:

where , . Denote the solution set of (24) by .

Lemma 8. Under Assumption 3(A1), is a convex function.

Proof. For any , , we have

where the first inequality uses Assumption 3(A1). The desired result follows.

Based on (20), combining (23) with Lemma 8, we can obtain the following conclusion.

Lemma 9. A point is a solution of (20) if and only if is a global optimal solution with the objective vanishing of (24).

In the following, we give the error bound for a polyhedral cone from [13] and error bound for a convex optimization from [14] to reach our aims.

Lemma 10. For polyhedral cone with , , , and , there exists a constant such that

Lemma 11. Let be a convex polyhedron in , and let be a convex quadratic function defined on . Let be the nonempty set of globally optimal solutions of the programming:

with being the optimal value of on . There exists a scalar such that

Before proceeding, we present the following definition introduced in [15].

Definition 12. The mapping is said to be strongly nonexpanding with a constant if .

By Lemma 8, is a convex function and the feasible set is a polyhedral. Combining this with Lemmas 10 and 11, we immediately obtain the following conclusion.

Theorem 13. Suppose that is -strongly -monotone with positive constants , , respectively, and is strongly nonexpanding with constant . Then, there exists constant such that

Proof. For any , let . Then, there exists such that . A direct computation yields that

where the second inequality follows from Definition 12 with constant , the third inequality follows from Definition 1(ii) with constants , , the fourth inequality follows from the Cauchy-Schwarz inequality, the fifth inequality follows from the fact that , for all , the sixth inequality follows from Lemma 11 with constant and Lemma 9, and the seventh inequality follows from Lemma 10 with constant . By (30) and letting , then the desired result follows.

Remark 14. It is clear that if is -strongly -monotone and is strongly nonexpanding, then

Moreover, the conditions which both and are Hölder continuous (or both and are Lipschitz continuous) in Theorem 13 are removed. Thus, Theorem 13 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 13 are also weaker than those in Theorem 3.1 in [15], Theorem 3.1 in [11], Theorem 3.1 in [10], and Theorem 2 in [9], respectively.

On the other hand, the condition that is -strongly -monotone and is strongly nonexpanding in Theorem 13 is extended compared with the condition that is strongly monotone with respect to (i.e., ) in Theorems 3.4 and 3.6 in [15], and it is also extended than compared with the condition is strongly monotone with respect to (i.e., ) in Theorem 3.1 in [11], and compared with the condition that , is strongly monotone (i.e., ) in Theorem 3.1 in [10].

Using the following Definition 15 developed from the complementarity conditions in (22), we can further detect the error bound of the GNCP.

Definition 15. A solution of the GNCP is said to be nondegenerate if it satisfies

Lemma 16. Suppose that Assumptions 3(A1) and 3(A2) hold, and the GNCP has a nondegenerate solution, say . Then,

where .

Proof. Since

by Assumption 3(A1), for any , we have

that is,

To prove the assertion, we only need to show that the solution set is equal to the set

For any , combining Lemma 9 with (20) yields that

Letting in (36) yields that

Since , using the similar technique to that of (21), we can obtain

where . Combining (39) with (40), we have .

On the other hand, for any , one has

Since , using the similar arguments to that of (21), one has

Combining this with (41) yields that

From (32), we deduce that

Thus, using (21), one has

Hence, .

Based on Lemma 16, we obtain the following conclusion.

Corollary 17. Suppose that the hypotheses of Lemma 16 hold. Then,

Theorem 18. Suppose that the hypotheses of Theorem 13 hold, and the GNCP has a nondegenerate solution. Then, there exists constant such that

Proof. For any , let . Then, there exists such that . Letting be a nondegenerate solution of GNCP and letting , , then

where the second equality uses the similar technique to that of (30), the third inequality follows from Corollary 17 and Lemma 10 with constant , and the last inequality is based on (36). By (48) and letting , the desired result follows.

In the following, we give an error bound of the Hölderian type [14].

Lemma 19. For , let be a convex quadratic function. If the set is nonempty, then there exist a positive integer (called the degree of singularity of the inequality system) and a positive scalar such that

where . Furthermore, if contains an interior point, then .

Based on (18) and (21), the GNCP can also be written as

From Lemma 19, we can establish the following global error bound for GNCP.

Theorem 20. Suppose that the hypotheses of Theorem 13 hold, and there exists point such that

Then, there exists constant such that

Proof. Let , where . By Lemma 8, we have is a convex quadratic function. Combining this with (51), using Lemma 19 with , this yields the following result

where is a positive constant.

Obviously, is a closed convex set. Thus, for any , there exists a vector such that

For convenience, we also let

From (50), we have , where is defined in (24), so for any , combining Lemma 10, one has

where is a positive constant, and the second and third 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. Thus,

Combining (56) with (59), for any , we have

where the second inequality follows from (56) with constant , the third inequality uses (59), the fifth inequality follows from (53), 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 , letting , then there exists such that , and a direct computation yields that

where the deduction of the second equality uses the similar technique to that of (30), and the third inequality is by (60). By (63) and letting , then the desired result follows.

Remark 21. When is strongly monotone with respect to , that is, , without the requirement of nondegenerate solution, the square root term in the error bound estimation is removed as stated in Theorem 20. Hence, the error estimation becomes more practical than that in Theorem 4.1 in [4].

3. Global Error Bound for the GLCP

In this section, we consider the linear case of the GCP such that mappings and are both linear; that is, , with , :

where

For problem (64), combining (18) with (23) and using a similar discussion in Lemmas 8 and 9, we also have the following conclusion.

Lemma 22. Under Assumption 3(A1), is a convex function.

Lemma 23. is a solution of the GLCP if and only if is global optimal solution with the objective vanishing of (64).

Based on (64), using the argument similar to that of Theorem 13, we can obtain the following conclusion.

Theorem 24. Under Assumptions 3(A1) and 3(A2), and that mappings and are both linear, there exists constant such that

Proof. For any , a direct computation yields that

where the first inequality follows from Lemma 11 with constant and Lemma 23, and the second inequality uses Lemma 10 with constant . By (67) and letting , the desired result follows.

Remark 25. Obviously, Assumption 3(A2) in Theorem 24 is weaker than Assumption (A2) in Theorem 4.1 in [4], Assumption 3(A1) coincides with Assumption (A1) in [4]. In addition, Theorem 24 is sharper than Theorem 4.1 in [4].

The following result further estimates the error bound for the GLCP.

Theorem 26. Suppose that the hypotheses of Theorem 24 hold, and the GLCP has a nondegenerate solution. Then, there exists constant such that

Proof. From Corollary 17, we have

where is a nondegenerate solution of GLCP, and is defined in (64). For any , a direct computation yields that

where the first inequality follows from Lemma 10 with constant , and the second inequality uses (36). Letting , the desired result follows.

Remark 27. The condition in Theorem 26 is weaker than that in Theorem 4.2 in [4].

Theorem 28. Suppose that the hypotheses of Theorem 24 hold, and there exists point such that (51) holds. Then there exists constant such that

Proof. Let , where . By Lemma 22, is a convex quadratic function, and is a closed convex set. For any , there exists a vector such that

Combining (51) and applying Lemma 19 yield the following result:

where is a positive constant. For convenience, we let

From (50), we have , where is defined in (64). So for any , combining Lemma 10 and using the similar technique to that of (56), one has

where is a positive constant.

Using the fact that

and using the similar technique to that of (57), one has

where the second inequality is by nonexpanding property of projection operator. Thus,

Combining (75) with (78), we know that for any , it holds that

where the second inequalities follows from (75) with constant , the third inequality follows from (78), the fifth inequality follows from (73), the sixth inequality follows by letting , and the seventh and ninth inequality follow from the fact that

By (79) and letting , the desired result follows.

Remark 29. In Theorem 28, without the requirement of nondegenerate solution, the square root term in the error bound estimation is removed. Hence, the error estimation becomes more practical than that in Theorem 4.1 in [4].

4. Comparison with Existing Error Bound

In the end of this paper, we will present an example to compare Theorem 13 and Theorem 2.5 in [5]. Furthermore, we will present two examples to show the conclusion in Theorem 13 can provide a global error bound for the GNCP, while the conclusion in Theorem 2.5 in [5] cannot do.

Example 30. When , (2) reduces to the generalized nonlinear complementarity problem of finding vector such that

For (81), using Theorem 13 with , we have

where .

Using Theorem 2.5 in [5] with , , and , we have that there exists constant such that

where . In addition,

In particular, when with constant , then there exists positive constant such that

In fact, we have

where the first inequality follows from the fact that , for all , the second inequality follows from the fact that , for all , the third inequality follows from the fact that , for all , the fourth inequality follows from the fact that , for all , and the fifth inequality follows from the Cauchy-Schwarz inequality.

Example 31. For mappings involved in problem (81), we set

It is easy to see that the solution set , and one has

where the first inequality follows from the fact that .

In fact, we consider the following four cases.

Case 1 ( and ). Then, and , and the desired result follows.

Case 2 ( and ). Then, and , and the desired result follows.

Case 3 ( and ). Then, and , and the desired result follows.

Case 4 ( and ). Then, and , and the desired result follows.

For any . By Theorem 13 with , we can obtain

as . Thus, Theorem 13 provides a global error bound for the GNCP. Using Theorem 2.5 in [5], for , we have

as . Thus, Theorem 2.5 in [5] fails in providing an error bound for this GNCP.

Example 32. For mappings involved in problem (81), we set

It is easy to see that the solution set . Without loss of generality, we let , and one has

where the inequality follows from the fact that

In fact, we consider the following four cases.

Case 1 (). Then, , and the desired result follows.

Case 2 (). Then, , and the desired result follows.

Case 3 ( and ). Then,

and the desired result follows.

Case 4 (, and ). Then,

and the desired result follows.

Thus, we obtain

For any , . By Theorem 13 with , we can obtain

as . Thus, Theorem 13 provides a global error bound for the GNCP.

On the other hand, using Theorem 2.5 in [5], for , we have

as , where is a constant with . Thus, Theorem 2.5 in [5] fails in providing an error bound for this GNCP.

5. Conclusion

In this paper, we established some global error bounds on the generalized nonlinear complementarity problems over a polyhedral cone, which improves the result obtained for variational inequalities and the GNCP [4, 5, 9–11] by weakening the assumptions. Surely, under milder conditions, we may establish global error bounds for GNCP and use the error bounds estimation to establish quick convergence rate of the methods for the GNCP. This is a topic for future research.

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 and 11101303), the Specialized Research Fund for the Doctoral Program of Chinese Higher Education (20113705110002), the Shandong Provincial Natural Science Foundation (ZR2010AL005), the Shandong Province Science and Technology Development Projects (2013GGA13034), and the Applied Mathematics Enhancement Program of Linyi University.

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