This paper deals with semidefinite complementarity problem (SDCP). Let denote the space of block-diagonal real matrices with m blocks of sizes . We endow with the inner product and norm:
where and denotes the matrix trace, is the Frobenius-norm of X and stands for the i-eigenvalue of X. Let denote the subspace comprising those that are symmetric, i.e., . We denote by the cone of symmetric positive semidefinite (positive definite) matrices in , We use the symbol to say that . To facilitate the presentation, let is the j-th block of , respectively. The SDCP is to find, for given mapping , an satisfying
The problem was firstly introduced in a slightly different form by Kojima, Shindoh and Hara  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    , 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. The concept of exceptional family of elements for a continuous function was first introduced by Smith  . Subsequently, Isac et al.  introdued a more general notion of exceptional family of elements. Using this notation, some existence theorems of a solution to nonlinear complementarity problems were presented in    . Zhao, Han et al. extended it to study existence conditions of a solution to variational inequality problems     . Recently, this notation was extended to study existence conditions of a solution to semidefinite complementarity problems and copositive cone complementarity problem    .
In this paper, Motivated by the previous researches, we discuss the nonemptyness and boundedness of the solution set for -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 -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.
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   .
Proposition 2.1 (Von Neumman-Theobald’s inequality) For any , it holds that , with equality if and only if and are simultaneously diagonalizable, where is the eigenvalue vector of and , respectively.
Proposition 2.2 Let , if , then and commute, i.e., and are simultaneously diagonalizable.
Proposition 2.3 (Fejer’s theorem) Let , it holds that for all if and only if . Moreover, for all if and only if .
Now, we present the definition and the property of -mapping and exceptional family of elements for SDCP on the subspace .
Definition 2.1 A mapping is said to be a -mapping, if there exists a nonnegative constant such that the following inequality holds for any distinct ,
where , and .
Definition 2.2  A sequence is
said to an exceptional family of elements for SDCP if and only if for any and every , there exists a real number such that
Theorem 2.1  If is a continuous mapping, then SDCP has either a solution or an exceptional family.
3. Main Result
To obtain our main results, we firstly present the following three lemmas in this section.
Lemma 3.1 If , is a matrix of size and , then there exists a subsequence such that has no a upper boundedness.
Proof. Suppose that the spectral decomposition of and is as follows, respectively.
where is the eigenvalue of , respectively. is the corresponding eigenvector, respectively. Noting that , we have that
In view of and , thus, there exists a
such that is unbounded. The above relation also show that there exists a
subsequence such that .
The next object is to show that there exists such that .
Assume that for any , one gets
Since A is a nonsingular, then we have . 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.
Lemma 3.2 If , and , then .
The proof of the following lemma is elementary, and omitted.
Lemma 3.3 If and , and is a cluster point of , then and .
Now, we present our main results as follows.
Theorem 3.1 If is a continuous mapping and there exists a strict feasible point for SDCP, i.e., , then the solution set of SDCP is nonempty.
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 , there exists a real number such that
Let . From the above first equation, one gets . Thus, for any , taking into account the above second equation and Proposition 2.3, we have
Denote by , . Obviously, and
When , we have that there exists a upper boundedness for from the formula (3.8).
When , one gets for sufficient large k. Noticing that , from Lemma 3.1, we have that for any , there exists a subsequence such that . Thus, . In view of the formula (3.8), one gets
which implies that
This is a contradiction with F being a -mapping. The proof is complete.
Theorem 3.2 If is a continuous -mapping and there exists a strict feasible point for SDCP, i.e., , then the solution set of SDCP is bounded.
Proof. Suppose that the solution set of SDCP is unbounded, i.e., there exists a
solution sequence such that . Obviously,
Noting that F is a 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
Noting that is bounded. Hence, there exists a subsequence such that
From Lemma 3.3, we have that .
On the other hand, from (3.16), one gets for any
Since , then
Obviously, . Thus, , which implies that from Lemma 3.2. This is a contradiction with . The proof is complete.
In this paper, the nonemptyness and boundedness of the solution set for -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.
This work is supported by National Natural Science Foundation of China (No.11761014, 11461015), Guangxi Natural Science Foundation (No. 2015GXNSFAA139010, 2017GXNSFAA198243) and Guangxi Colleges and Universities Key Laboratory of Mathematical and Statistical Model (No. 2016GXKLMS010), Guangxi basic ability improvement project for the Middle-aged and young teachers of colleges and universities (2017KY0068, KY2016YB069).
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