Journal of Applied Mathematics

Volume 2012 (2012), Article ID 219478, 15 pages

doi:10.1155/2012/219478

Research Article

The Solution Set Characterization and Error Bound for the Extended Mixed Linear Complementarity Problem

Hongchun Sun¹ and Yiju Wang²

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

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

Received 19 September 2012; Accepted 8 December 2012

Academic Editor: Jian-Wen Peng

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.

For the extended mixed linear complementarity problem (EML CP), we first present the characterization of the solution set for the EMLCP. Based on this, its global error bound is also established under milder conditions. The results obtained in this paper can be taken as an extension for the classical linear complementarity problems.

1. Introduction

We consider that the extended mixed linear complementarity problem, abbreviated as EMLCP, is to find vector such that

where , , , , ,, , , , . We assume that the solution set of the EMLCP is nonempty throughout this paper.

The EMLCP is a direct generalization of the classical linear complementarity problem and a special case of the generalized nonlinear complementarity problem which was discussed in the literature ([1, 2]). The extended complementarity problem plays a significant role in economics, engineering, and operation research, and so forth [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 [4].

Up to now, the issues of the solution set characterization and numerical methods for the classical linear complementarity problem or the classical nonlinear complementarity problem were fully discussed in the literature (e.g., [5–8]). On the other hand, the global error bound is also an important tool in the theoretical analysis and numerical treatment for variational inequalities, nonlinear complementarity problems, and other related optimization problems [9]. The error bound estimation for the classical linear complementarity problems (LCP) was fully analyzed (e.g., [7–12]).

Obviously, the EMLCP is an extension of the LCP, and this motivates us to extend the solution set characterization and error bound estimation results of the LCP to the EMLCP. To this end, we first detect the solution set characterization of the EMLCP under milder conditions in Section 2. Based on these, we establish the global error bound estimation for the EMLCP in Section 3. These constitute what can be taken as an extension of those for linear complementarity problems.

We end this section with 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 . We use to denote the nonnegative orthant in and use and to denote the vectors composed by elements , respectively. For simplicity, we use for column vector . We also use to denote a nonnegative vector if there is no confusion.

2. The Solution Set Characterization for EMLCP

In this section, we will characterize the solution set of the EMLCP. First, we can give the needed assumptions for our analysis.

Assumption 2.1. For the matrices involved in the EMLCP, we assume that the matrix is positive semidefinite.

Theorem 2.2. Suppose that Assumption 2.1 holds; the following conclusions hold.

(i) If is a solution of the EMLCP, then

where , , , , and denotes the solution set of EMLCP.

(ii) If and are two solutions of the EMLCP, then

(iii) The solution set of EMLCP is convex.

Proof. Set

For any , since , we have

Since , using the similar arguments to that in (2.4), we have

Combining (2.4) with (2.5), one has

By (2.6), we have

By Assumption 2.1, one has

Combining (2.7) with (2.8), we have

That is,

Using again, we have

Using again, using the similar arguments to that in (2.11), we have

From (2.9), (2.4), and (2.11), one has

Combining (2.5) with (2.12) yields

Combining this with (2.13) yields

From (2.10) and (2.15), one has

By (2.10) and (2.16), we obtain that follows.

On the other hand, for any , then , and

and one has

Using (2.18), one has

Thus, we have that .

Since and are two solutions of the EMLCP, by Theorem 2.2  , we have

Combining this with , one has

On the other hand, from , we can deduce

From (2.21) and (2.22), thus, we have that Theorem 2.2   holds.

If solution set of the EMLCP is single point set, then it is obviously convex. In this following, we suppose that and are two solutions of the EMLCP. By Theorem 2.2  , we have

For the vector , by (2.23), we have

Using the similar arguments to that in (2.24), we can also obtain

Combining (2.24) and (2.25) with the conclusion of Theorem 2.2 (i), we obtain the desired result.

Corollary 2.3. Suppose that Assumption 2.1 holds. Then, the solution set for EMLCP has the following characterization:

Proof. Set

For any , then , combining this with . Using the similar arguments to that in (2.5) and (2.12), we have

Combining this with , one has

From , we have

Thus, by Theorem 2.2, one has .

On the other hand, for any , by Theorem 2.2  , we have , , and , that is,

Thus, .

Using the following definition developed from EMLCP, we can further detect the solution structure of the EMLCP.

Definition 2.4. A solution of the EMLCP is said to be nondegenerate if it satisfies

Theorem 2.5. Suppose that Assumption 2.1 holds, and the EMLCP has a nondegenerate solution, say . Then, the following conclusions hold.

(i) The solution set of EMLCP

(ii) If the matrices and are the full-column rank, where , , then is the unique nondegenerate solution of EMLCP.

Proof. Set

From Corollary 2.3, one has . In this following, we will show that . For any , then , combining this with . Using the similar arguments to that in (2.14), we have

Combining this with , one has

Combining with (2.36), one has

Since is a nondegenerate solution, combining this with (2.37), we have . That is, .

Let be any nondegenerate solution. Since is a nondegenerate solution, then we have

Combining (2.38) with (2.39), we have

If , then by (2.39). By (2.38) again, we can deduce that

On the other hand, for the and which are solutions of EMLCP, and combining Theorem 2.2  , we have . Using , we can deduce that

Combining Theorem 2.2   again, we also have

For any , that is, , and combining (2.43), we obtain

Combining this with the fact that , we can deduce that

From (2.41) and (2.42), we obtain

Thus, by the full-column rank assumption on . Using , combining (2.40) with (2.44), we can deduce that

That is, by the full-column rank assumption on . Thus, the desired result follows.

The solution set characterization obtained in Theorem 2.2   coincides with that of Lemma  2.1 in [7], and the solution set characterization obtained in Theorem 2.5   coincides with that of Lemma  2.2 in [8] for the linear complementarity problem.

3. Global Error Bound for the EMLCP

In this following, we will present a global error bound for the EMLCP based on the results obtained in Corollary 2.3 and Theorem 2.5  . Firstly, we can give the needed error bound for a polyhedral cone from [13] and following technical lemmas to reach our claims.

Lemma 3.1. For polyhedral cone with , , and , there exists a constant such that

Lemma 3.2. Suppose that is a solution of EMLCP, and let

then, there exists a constant , such that for any , one has

Proof. Similar to the proof of (2.14), we can obtain

We consider the following linear programming problems

From the assumption, we know that is an optimal point of the linear programming problem. Thus, there exist optimal Lagrange multipliers , , and such that

From (3.6), we can easily deduce that

Thus, for any , from the first equation in (3.6), we have

Where is a constant. Let , then the desired result follows.

Now, we are at the position to state our results.

Theorem 3.3. Suppose that Assumption 2.1 holds. Then, there exists a constant such that for any , there exists such that

where

Proof. Using Corollary 2.3 and Lemma 3.1, there exists a constant , for any , and there exists such that

Where is a solution of EMLCP. Now, we consider the right-hand-side of expression (3.11).

Firstly, by Assumption 2.1, we obtain that

is a convex function. For any , we have

Combining this with , we can deduce that

Secondly, we consider the last item in (3.11). By Assumption 2.1, there exists a constant such that for any ,

where the first equality is based on the Taylor expansion of function on point, the second inequality follows from the fact that is a solution of EMLCP and the fact that for any , and the last inequality is based on Lemma 3.2. By (3.11)–(3.15), we have that (3.9) holds.

The error bound obtained in Theorem 3.3 coincides with that of Theorem  2.4 in [11] for the linear complementarity problem, and it is also an extension of Theorem  2.7 in [7] and Corollary  2 in [14].

Theorem 3.4. Suppose that the assumption of Theorem 2.5 holds. Then, there exists a constant , such that for any , there exists a solution such that

where is defined in Theorem 3.3.

Proof. From Theorem 2.5, using the proof technique is similar to that of Theorem 3.3. For any , there exist and a constant such that

Combining this with (3.14), we can deduce that (3.16) holds.

4. Conclusion

In this paper, we presented the solution Characterization, and also established global error bounds on the extended mixed linear complementarity problems which are the extensions of those for the classical linear complementarity problems. Surely, we may use the error bound estimation to establish quick convergence rate of the noninterior path following method for solving the EMLCP just as was done in [14], and this is a topic for future research.

Acknowledgments

This work was supported by the Natural Science Foundation of China (Grant no. 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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