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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

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 ([

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., [

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

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

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

For the matrices

Suppose that Assumption

If

If

The solution set of EMLCP is convex.

For any

On the other hand, for any

Suppose that Assumption

Set

On the other hand, for any

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

A solution

Suppose that Assumption

The solution set of EMLCP

If the matrices

The solution set characterization obtained in Theorem

In this following, we will present a global error bound for the EMLCP based on the results obtained in Corollary

For polyhedral cone

Suppose that

Similar to the proof of (

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

Suppose that Assumption

Using Corollary

Firstly, by Assumption

Secondly, we consider the last item in (

The error bound obtained in Theorem

Suppose that the assumption of Theorem

From Theorem

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 [

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).