Variational Inequalities and Vector OptimizationView this Special Issue
The Solution Set Characterization and Error Bound for the Extended Mixed Linear Complementarity Problem
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.
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 . 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 .
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 . 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.
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:
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.
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 , and the solution set characterization obtained in Theorem 2.5 coincides with that of Lemma 2.2 in  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  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.
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 , and this is a topic for future research.
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).
F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequality and Complementarity Problems, Springer, New York, NY, USA, 2003.
L. Walras, Elements of Pure Economics, George Allen and Unwin, London, UK, 1954.