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This paper investigates the set-valued complementarity problems (SVCP)
which poses rather different features from those that classical complementarity problems
hold, due to tthe fact that he index set is not fixed, but dependent on

The

The SVNCP

This corresponds to

where

Besides the above various complementarity problems, SVNCP

Let

Note first that for each

From all the above, we have seen that SVNCP

A few words about the notations used throughout the paper. For any

It is well known that various matrix classes paly different roles in the theory of linear complementarity problem, such as

A matrix

Note that

Let

If

We point out that the set-valued mapping

If

We only need to show (

There is another point worthy of pointing out. We mentioned that the classical linear complementarity problem LCP

Consider the set-valued linear complementarity problem SVLCP

Let

if taking

if taking

Repeating the above process yields a sequence

A matrix

From [

Let

For

A matrix

For the classical linear complementarity problem, we know that

The set of matrices

strongly semimonotone if for any nonzero

weakly semimonotone if for any nonzero

Unlike the classical linear complementarity problem case, here are parallel results regarding set-valued linear complementarity problem which strong (weak) semi-monotonicity plays in.

For the SVLCP

If the set of matrices

If SVLCP

(a) It is clear that, for any positive mapping

(b) Suppose

Theorem

Let

For any nonzero

So far, we have seen some major difference between the classical complementarity problem and set-valued complementarity problem. Such phenomenon undoubtedly confirms that it is an interesting, important, and challenging task to study the set-valued complementarity problem, which, to some extent, is the main motivation of this paper.

To close this section, we introduce some other concepts which will be used later too. A function

Recently, many authors study other classes of complementarity problems, in which another type of vector

SOL(

SOL(

SOL

SOL

Besides, for the purpose of comparison, we restrict that

It is easy to see that the solution set of SINCP

Let

However, the solution set of SVNCP

Let

Let

Let

Let

Similarly, Example

Let

Let

In spite of these, we obtain some results which describe the relationship among them.

Let

Parts (a) and (b) follow immediately from the fact

For further characterizing the solution sets, we recall that for a set-valued mapping

For SVNCP(

In fact, the desired result follows from

It is well known that one of the important approaches for solving the complementarity problems is to transfer it to a system of equations or an unconstrained optimization via NCP functions or merit functions. Hence, we turn our attention in this section to address merit functions for SVNCP

A function

Assume that there exists a set

Noticing that if

One may ask when condition (

For simplification of notations, we write

For SVLCP

In the case of a linear complementarity problem, that is,

For SVLCP

We argue this result by contradiction. Suppose there exists a sequence

Note that the condition (

For SVLCP

Suppose that there exist a nonzero vector

The above conclusion is equivalent to say that for each

If

Noting that the problem (

The foregoing result indicates that the set-valued complementarity problem is different from the classical complementarity problem, since it restricts that some components of the solution must be positive or zero, which is not required in the classical complementarity problems.

Moreover, the set-valued complementarity problem can be further reformulated to be an equation, that is, finding

Suppose that

In this paper, we have paid much attention to the set-valued complementarity problems which posses rather different features from those of classical complementarity problems. As suggested by one referee, we here briefly discuss the relation between stochastic variational inequalities and the set-valued complementarity problems. Given

Due to some major difference between set-valued complementarity problems and classical complementarity problems, there are still many interesting, important, and challenging questions for further investigation as below, to name a few.

How to extend other important concepts used in classical linear complementarity problems to set-valued cases (like

How to propose an effective algorithm to solve (

Can we provide some sufficient conditions to ensure the existence of solutions? One possible direction is to use

The authors would like to thank three referees for their carefully reading and suggestions which help to improve this paper. J. Zhou is supported by National Natural Science Foundation of China (11101248, 11271233), Shandong Province Natural Science Foundation (ZR2010AQ026, ZR2012AM016), and Young Teacher Support Program of Shandong University of Technology. J. S. Chen is supported by National Science Council of Taiwan. G. M. Lee is supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (no. 2012-0006236).

_{0}matrix linear complementarity problems