Some Properties of Solution to Semidefinite Complementarity Problem

In this paper, we discuss the nonemptyness and boundedness of the solution set for * P -semidefinite complementarity problem by using the concept of exceptional family of elements for complementarity problems over the cone of semidefinite matrices, and obtain a main result that if the corresponding problem has a strict feasible point, then its solution set is nonemptyness and boundedness.


Introduction
This paper deals with semidefinite complementarity problem (SDCP).Let χ denote the space of n n × block-diagonal real matrices with m blocks of sizes ( ) , , , We endow χ with the inner product and norm: ( ) where ,

X Y χ ∈ and [ ]
tr ⋅ denotes the matrix trace, X is the Frobenius-norm of X and ( ) i X λ stands for the i-eigenvalue of X.Let  denote the subspace comprising those X χ ∈ that are symmetric, i.e., T X X = .We denote by X Y is the j-th block of , X Y χ ∈ , respectively.The SDCP is to find, for given mapping : F →   , an ( ) The problem was firstly introduced in a slightly different form by Kojima, Shindoh and Hara [1] as a model unifying various problems arising from system and control theory and combinatorial optimization.It can be regarded as a generalization of standard complementarity problem (CP).
Recently, there has been growing interest in searching for solutions methods for SDCP [1] [2] [3], but the assumption that SDCP has a solution is necessary for these solutions methods.It follows that the research of solution conditions for SDCP has played a very important role in both theory and practical applications.Among them, the concept of exceptional family is a powerful tool to study existence of the solution to CP.In this paper, Motivated by the previous researches, we discuss the nonemptyness and boundedness of the solution set for * P -semidefinite complementarity problem by using the concept of exceptional family of elements for complementarity problems over the cone of semidefinite matrices, and we prove that if the corresponding problem has a strict feasible point, then its solution set is nonemptyness and boundedness.
The remainder of this paper is organized as follows.The preliminary results which will be used in this paper are stated in Section 2. In Section 3, we discuss the nonemptyness and boundedness of the solution set for * P -semidefinite complementarity problem by using the concept of exceptional family of elements for complementarity problems over the cone of semidefinite matrices.
Conclusions are drawn in Section 4.

Preliminaries
In this section, we firstly recall some matrix properties that we shall employ throughout this paper.Their proofs and mores details can be found for instance in [15] Now, we present the definition and the property of * P -mapping and exceptional family of elements for SDCP on the subspace  .
Definition 2.1 A mapping : F →   is said to be a * P -mapping, if there exists a nonnegative constant γ such that the following inequality holds for any distinct , where said to an exceptional family of elements for SDCP if and only if for any r and every i I ∈ , there exists a real number 0 Theorem 2.1 [12] If : F →   is a continuous mapping, then SDCP has either a solution or an exceptional family.

Main Result
To obtain our main results, we firstly present the following three lemmas in this section.
X C has no a upper boundedness.
Proof.Suppose that the spectral decomposition of r i X and i C is as follows, respectively.
T T 1 1 , , where , r j l λ γ is the eigenvalue of , r i i X C , respectively., r j l ξ η is the corresponding eigenvector, respectively.Noting that 0, 0 , we have that 0, 0. ξ η = for any l , one gets Since A is a nonsingular, then we have 0 0 r j ξ = .This is a contradiction.
Combining the above relations, we have ( ) .
The proof is complete.
From Proposition 2.1 and Proposition 2.2, we can get the following lemma.
The proof of the following lemma is elementary, and omitted.Proof.Suppose that there exists no solution for SDCP, then from Theorem 2.1, we have that there exists an exceptional family of elements for SDCP, and for every i I ∈ , there exists a real number 0 From the above first equation, one gets 0 Thus, for any i I ∈ , taking into account the above second equation and Proposition 2.3, we have Denote by When 2 i I ∈ , we have that there exists a upper boundedness for When , from Lemma 3.1, we have that for any In view of the formula (3.8), one gets ( ) ( ) which implies that ( ) ( ) This is a contradiction with F being a * P -mapping.The proof is complete.
Noting that F is a * P mapping, we have that for any k, ( ) From the formula (3.12), one gets Taking into account Proposition 2.3 and the formula (3.15), we get Obviously, ( ) 0 , 0 F X X ≥ .Thus, ( ) 0 , 0 F X X = , which implies that 0 X = from Lemma 3.2.This is a contradiction with 1 X = .The proof is complete.

Conclusion
In this paper, the nonemptyness and boundedness of the solution set for * P -semidefinite complementarity problem have been discussed by using the concept of exceptional family of elements for complementarity problems over the cone of semidefinite matrices, and a main result has been shown that if the corresponding problem has a strict feasible point, then its solution set is nonemptyness and boundedness.
facilitate the presentation, let , j j thus, there exists a 0 j such that { } 0 r j λ is unbounded.The above relation also show that there exists a Y. Chen et al.
present our main results as follows.Theorem 3.1 If : F →   is a continuous * P mapping and there exists a strict feasible point for SDCP, i.e., solution set of SDCP is nonempty.
there exists a subsequence { } hand, from (3.16), one gets for any n k [4] concept of exceptional family of elements for a continuous function was first introduced by Smith[4].