On Set-Valued Complementarity Problems

and Applied Analysis 3 Proof. Note first that for each x the inner problem ψ (x) := max

Besides the above various complementarity problems, SVNCP(, Ω) has a close relation with the Quasi-variational inequality, a special of the extended general variational inequalities [6,7], and min-max programming, which is elaborated as below.
(vi) Min-max programming [8], which is to solve the following problem: min where  : R  × Ω → R is a continuously differentiable function, and Ω is a compact subset in R  .First, we define () := max ∈Ω (, ).Although  is not necessarily Frechet differentiable, it is directional differentiable (even semi-smooth), see [9].Now, let us check the first-order necessary conditions for problem (15).In fact, if  * is a local minimizer of (15), then which is equivalent to inf where Ω() means the active set at , that is, Ω() := { ∈ Ω|() = (, )}.At our first glance, the formula ( 17) is not related to SVNCP(, Ω).Nonetheless, we will show that if Ω is convex and the function (, ⋅) is concave over Ω, then the first-order necessary conditions form an SVNCP(, Ω), see below proposition.

Proposition 1.
Let Ω be nonempty, compact, and convex set in R  .Suppose that, for each , the function (, ⋅) is concave over Ω.If  * is a local optimal solution of (15), then there exists  * ∈ Ω( * ), such that Abstract and Applied Analysis 3 Proof.Note first that for each  the inner problem is a concave optimization problem, since (, ⋅) is concave and Ω is convex.This ensures that Ω(), which denotes the optimal solution set of (19), is convex as well.Now we claim that the function is concave over Ω( * ).Indeed, for  1 ,  2 ∈ Ω( * ) and  ∈ [0, 1], we have where we use the fact that  inf Hence, for arbitrary  > 0, we can find   ∈ Ω( * ), such that inf that is, In particular, plugging in  = 0 in (24) implies Since Ω is bounded and Ω( * ) is closed, we can assume, without loss of generality, that   →  * ∈ Ω( * ) as  → 0. From all the above, we have seen that SVNCP(, Ω) given as in (2) covers a range of optimization problems.Therefore, in this paper, we mainly focus on SVNCP(, Ω).Due to its equivalence to SVCP (1), our analysis and results for SVNCP(, Ω) can be carried over to SVCP (1).This paper is organized as follows.In Section 1, connection between SVNCP(, Ω) and various optimization problems is introduced.We recall some background materials in Section 2.Besides comparing the set-valued complementarity problems with the classical complementarity problems, we analyze the solution set of SVCP in Section 3.Moreover, properties of merit functions for SVCP are studied in Section 4, such as level bounded and error bound.Finally, some possible research directions are discussed.

Focus on SVLCP(𝑀, 𝑞, Ω)
It is well known that various matrix classes paly different roles in the theory of linear complementarity problem, such as matrix, -matrix, -matrix, and -matrix, see [1,12] for more details.Here we recall some of them which will be needed in the subsequent analysis.
Note that ∈R × is an -matrix, if and only if the classical linear complementarity problem LCP(, ) is feasible for all  ∈ R  , see [12,Prop. 3.1.5].Moreover, the above condition in Definition 2 is equivalent to see [13,Remark 2.2].However, such equivalence fails to hold for its corresponding cases in set-valued complementarity problem.In other words, is not equivalent to It is clear that (34) implies (35).But, the converse implication does not hold, which is illustrated in Example 3.
There is another point worthy of pointing out.We mentioned that the classical linear complementarity problem LCP(, ) is feasible for all  ∈ R  if and only if  ∈ R × is a -matrix; that is, there exists  ∈ R  , such that  > 0,  > 0. (44) Is there any analogous result in the set-valued set?Yes, we have an answer for it in Theorem 5 below.
Proof.Let  be any mapping from R  to R  being bounded from below, that is, there exists  ∈ R, such that () ≥ .
Example 7 shows that the aforementioned implication is not valid as well in set-valued complementarity problem.
For the classical linear complementarity problem, we know that  is semimonotone if and only if LCP(, ) with  > 0 has a unique solution (zero solution), see [12,Theorem 3.9.3].One may wonder whether such fact still holds in set-valued case.Before answering it, we need to know how to generalize concept of semi-monotonicity to its corresponding definition in the set-valued case.
Theorem 10(b) says that the weak semi-monotonicity is a necessary condition for zero being the unique solution of SVLCP(, , Ω).However, it is not the sufficient condition, see Example 11.We also notice that the set-valued mapping Ω() is even continuous in Example 11.
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  : R  → R is level bounded, if the level set { | () ≤ } is bounded for all  ∈ R. The metric projection of  to a closed convex subset  ⊂ R  is denoted by Π  (), that is, Π  () := arg min ∈ ‖ − ‖.The distance function is defined as dist(, ) := ‖ − Π  ()‖.
These two problems are denoted by NCP( min ) and NCP( max ), respectively.Is there any relationship among their solutions sets?In order to further describing such relationship, we adapt the following notations: Besides, for the purpose of comparison, we restrict that Ω() is fixed; that is, there exists a subset Ω in R  , such that Ω() = Ω for all  ∈ R  .It is easy to see that the solution set of SINCP(, Ω) is ⋂ ∈Ω SOL(  ), but that of SVNCP(, Ω) is ⋃ ∈Ω SOL(  ), where   ():=(, ).Hence, the solution set of SINCP(, Ω) is included in that of SVNCP(, Ω).In other words, we have ŜOL (, Ω) ⊆ SOL (, Ω) . (67) The inclusion (67) can be strict as shown in Example 12.In spite of these, we obtain some results which describe the relationship among them.Proof.Parts (a) and (b) follow immediately from the fact Part (c) is from (67), since the two sets in the left side of (c) is ŜOL(, Ω) by [18].
For further characterizing the solution sets, we recall that for a set-valued mapping  : R   R  , its inverse mapping (see [9,Chapter 5]) is defined as Theorem 17.For SVNCP(, Ω), we have where the second equality is due to the definition of inverse mapping given as above.

Merit Functions for SVNCP and SVLCP
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(, Ω) and SVLCP(, , Ω).
For example, the natural residual  NR (, ) = min{, } and the Fischer-Burmeister function  FB (, ) = √  2 +  2 −(+) are popular NCP-functions.Please also refer to [19] for a detailed survey on the existing NCP-functions.In addition, a real-valued function  : R  → R is called a merit (or residual) function for a complementarity problem if () ≥ 0 for all  ∈ R  and () = 0 if and only if  is a solution of the complementarity problem.Given an NCP-function , we define  (, ) := ‖Φ (,  (, ))‖ where Φ (, ) := ( ( 1 ,  1 ) , . . .,  (  ,   )) . ( Then, it is not hard to verify that the function given by  () := min is a merit function for SVNCP(, Ω).Note that the merit function (74) is rather different from the traditional one, because the index set is not a fixed set, but dependent on .
We say that a merit function () has a global error bound with a modulus For more information about the error bound, please see [20] which is an excellent survey paper regarding the issue of error bounds.
Theorem 18. Assume that there exists a set Ω ⊂ R  , such that Ω() = Ω for all  ∈ R  , and that for each  ∈ Ω, (, ) is a global error bound of (  ) with the modulus () > 0, that is, In addition, if then () = min ∈Ω (, ) provides a global error bound for SVNCP(, Ω) with the modulus .
Proof.Noticing that if Ω() = Ω for all  ∈ R  , then It then follows from Theorem 17 that Thus, the proof is complete.
One may ask when condition (77) is satisfied?Indeed, the condition (77) is satisfied if (i) Ω is a finite set; (ii) (, ) = () + () where () is continuous, and for each  ∈ Ω the matrix () is a -matrix.In this case the modulus () takes an explicitly formula, that is, see [21,22].Hence, we see that is well defined because () is continuous, and Ω is compact.
Proof.We argue this result by contradiction.Suppose there exists a sequence {  } satisfying ‖  ‖ → ∞, and (  ) is bounded.Then,
Note that the condition (84) is equivalent to which is also equivalent to saying that each matrix () for  ∈ lim sup  → ∞ Ω() is a  0 -matrix.(89) Proof.Suppose that there exist a nonzero vector  0 , and  0 ∈ ∩ Ñ∈ ∞ ∪ ∈ ÑΩ( 0 ), such that Similar to the argument as in Theorem where the inequality in the latter case comes from the fact that   ( 0 ) ≤   (( 0 ) 0 )  +   ( 0 ) < 0. Thus, This contradicts the level boundedness of () since    0 → ∞.
where Θ  : R  → R + Then, the desired result follows.
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.

Further Discussions
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  : R  × Ξ → R,   ⊂ R  and Ξ ⊂ R  , a set representing future states of knowledge, the stochastic variational inequalities is to find  ∈   , such that or equivalently there exists a subset Ξ 0 ⊂ Ξ with (Ξ 0 ) = 0, such that  ≥ 0,  (, ) ≥ 0,    (, ) = 0, ∀ ∈ Ξ \ Ξ 0 .
Hence the stochastic complementarity problem is, in certain extent, a semi-infinite complementarity problem (SICP).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.