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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 130682, 14 pages
Lipschitz Continuity of the Solution Mapping of Symmetric Cone Complementarity Problems
1Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China
2Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan
3Mathematics Division, National Center for Theoretical Sciences (Taipei Office), Taipei 10617, Taiwan
Received 24 February 2012; Accepted 25 August 2012
Academic Editor: Malisa R. Zizovic
Copyright © 2012 Xin-He Miao and Jein-Shan Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper investigates the Lipschitz continuity of the solution mapping of symmetric cone (linear or nonlinear) complementarity problems (SCLCP or SCCP, resp.) over Euclidean Jordan algebras. We show that if the transformation has uniform Cartesian P-property, then the solution mapping of the SCCP is Lipschitz continuous. Moreover, we establish that the monotonicity of mapping and the Lipschitz continuity of solutions of the SCLCP imply ultra P-property, which is a concept recently developed for linear transformations on Euclidean Jordan algebra. For a Lyapunov transformation, we prove that the strong monotonicity property, the ultra P-property, the Cartesian P-property, and the Lipschitz continuity of the solutions are all equivalent to each other.
Let (we use for short in subsequent content) be a Euclidean Jordan algebra and be the symmetric cone in . Given a continuous transformation and , the symmetric cone complementarity problem denoted by SCCP is to find a vector such that When reduces to a linear transformation , the above problem is called the symmetric cone linear complementarity problem and is denoted by SCLCP, that is, the symmetric cone linear complementarity problem is to find a vector such that These classes of symmetric cone complementarity problems provide a unified framework for the linear or nonlinear complementarity problems (LCP or NCP, resp.) over the nonnegative orthant cone in , that is, and (see [1–4]), the second-order cone (linear or nonlinear) complementarity problems (SOCLCP or SOCCP, resp.), that is, and (see [5–8]), and the semidefinite (linear or nonlinear) complementarity problems (SDLCP or SDCP, resp.), that is, and (see [9–12]). It is also known that the complementarity problem is special case of variational inequality problem which has a wide range of applications, see [3, 9].
One of the important issues in complementarity problems is to characterize the Lipschitz continuity of its solutions (or called the Lipschitz continuity of solution mapping) with respect to . For , let be the set of all solutions to SCCP. Then, we intend to know under what conditions the multivalued solution mapping of SCCP is Lipschitz continuous. In other words, under what conditions, there will exist such that for all , satisfying and , where is the closed unit ball in . That is, if there exists such that Note that the Lipschitz constant depends only on the continuous transformation . Below is a brief history regarding this issue. For LCP, it is well known that the Lipschitz continuity of the solution mapping with respect to can be described in any one of the following ways:(i)the matrix is -matrix (see [13, 14]);(ii)LCP has a unique solution for all (i.e., GUS-property of );(iii)for any , the solution set and the set-valued mapping are Lipschitzian.In particular, Mangasarian and Shiau  showed that if is a -matrix, then solutions of linear inequalities, programs, and LCP are Lipschitz continuous. Murthy et al.  showed that is a -matrix if and only if the LCP has a solution for all and the solution mapping is Lipschitzian. Gowda and Sznajder  generalized the above result to affine variational inequality problems, while Yen  studied Lipschitz continuity of the solution mapping of variational inequalities with a parametric polyhedral constraint. As for NCP, Levy  obtained that the solution mapping is locally single-valued and Lipschitz continuous under suitable conditions. How about when is nonpolyhedral? Balaji et al.  proved that being monotone and the Lipschitz continuity of the solution mapping of SDLCP imply the GUS-property, while Chen and Qi in  employed Cartesian -property to guarantee the GUS-property and the locally Lipschitzian property of the solution mapping of SDLCP. These make a complete extension of (i)–(iii) to their counterparts in SDLCP. A natural question arises here: can the above results be extended to a general symmetric cone case which is a unified framework?
In fact, there has been some papers dealing with the SCLCP over Euclidean Jordan algebras. For example, Balaji  established the result that if has the Lipschitzian -property, then has the positive principal minor property. Gowda et al.  showed that if has -property, then SCLCP has a nonempty compact set for all . In addition, Tao and Gowda  used degree-theoretic arguments to show that under a certain -type condition, every symmetric cone nonlinear complementarity problem SCCP has a solution. However, it still remains open under what conditions the solution map of SCCP is Lipschitz continuous. In this paper, we explore new results regarding Lipschitz continuity of the solution mapping of the SCLCP or SCCP over Euclidean Jordan algebras. In Theorem 3.1, we show that if the transformation has the uniform Cartesian -property with modulus , then the solution mapping is Lipschitz continuous with respect to . Meanwhile, we give examples to show that the solution mapping of nonstrong monotone SCLCP is not Lipschitz continuous with respect to , and GUS-property does not imply Lipschitz continuity of the solution mapping.
On the other hand, various -properties and GUS-property have been investigated in the literature [4, 9, 10, 13, 16, 19, 21–24]. Relations among them are well studied as well. In [19, Theorem 2.2], it is proved that if the linear transformation in SDLCP has the monotonicity property and is Lipschitzian, then has the -property and the GUS-property. The concept of -property in was extended to a general Euclidean Jordan algebra, called ultra -property . Hence, it is desirable to know whether [19, Theorem 2.2] can be true or not in SCLCP if -property is replaced by ultra -property. In this paper, we answer this question positively, see Theorem 3.8. Further, for the Lyapunov transformation , we present several equivalent conditions for the ultra -property of .
Next are a few words about notations and some basic concepts employed. For a vector , the norm is denoted by , where denotes the Euclidean inner product. For the Euclidean Jordan algebra , let denote the set of all continuous linear transformation , and denote the set of all (invertible) linear transformations such that . For the convex set , let denote the interior of the . means the adjoint operator of . The identical transformation on will be denoted by . For the SCCP, the solution set of SCCP is denoted by . For the SCLCP, the solution set of SCLCP is denoted by SOL or .
An Euclidean Jordan algebra is a triple ( for short), where is a finite-dimensional inner product over and is a bilinear mapping satisfying the following three conditions:(i) for all , ; (ii) for all , , where ; (iii) for all , , . We call the Jordan product of and . In addition, if there is an element such that for all , the element is called the identity element in . In a given Euclidean Jordan algebra , the set of squares is a symmetric cone [25, Theorem III.2.1]. In other words, is a self-dual closed convex cone, and, for any two elements , , there exists an invertible linear transformation such that and . For any , we write An element such that is called an idempotent in ; it is a primitive idempotent if it is nonzero and cannot be written as a sum of two nonzero idempotents. We say that a finite set of primitive idempotents in is a Jordan frame if where is called the rank of . Now, we recall the spectral and Peirce decompositions of an element in .
Theorem 2.1 ((spectral decomposition) [25, Theorem III.1.2]). Let be an Euclidean Jordan algebra. Then, there is a number such that for every , there exists a Jordan frame and real numbers with Here, the numbers for are the eigenvalues of and the expression is the spectral decomposition (or the spectral expansion) of .
In an Euclidean Jordan algebra , corresponding to the convex cone , let denote the metric projection onto , namely, for an , if and only if and for all . It is well known that is unique. For any , combining the spectral decomposition of with the metric projection of onto , we have the expression of metric projection as follows (see ):
The Peirce Decomposition
Fix a Jordan frame in an Euclidean Jordan algebra . For , we define the following eigenspaces:
Theorem 2.2 (see [25, Theorem IV.2.1]). The space is the orthogonal direct sum of spaces . Furthermore, Hence, given any Jordan frame , we can write any element as where and . The expression is called the Peirce decomposition of .
Next, we recall concept of Lyapunov transformation and its relevant conclusions which will be used in our analysis later. In an Euclidean Jordan algebra , for any , we define the corresponding Lyapunov transformation by for any . As remarked in [21, page 209], traditionally, the notation has been used the Lyapunov transformation . As employed in , we also reserve the notation for the Lyapunov transformation and write to denote the image of an element under a linear transformation . We say that elements and operator commute if . It is well known that and operator commute if and only if and have their spectral decompositions with respect to a common Jordan frame [25, Lemma X.2.2].
Property 1 (see [21, Proposition 6]). For , , the following conditions are equivalent:(a), , and ;(b), , and . Moreover, in this case, elements and operator commute. That is, and have their spectral decompositions with respect to a common Jordan frame.
In fact, from Property 1 and definition of (1.1), it can be seen that SCCP is equivalent to find a such that In addition, if is a solution of SCCP, then and operator commute. Now, we review various monotonicity and -property for a continuous transformation .
Definition 2.3. Let be an Euclidean Jordan algebra. A continuous transformation is said to be(a)monotone if , for all , ;(b)strictly monotone if , for all ;(c)strongly monotone if there is such that
It is said to have(d) GUS-property if SCCP has a unique solution for any ;(e)-property if (f)-property if for any .
Remark 2.4. (i) When is linear, strict monotonicity and strong monotonicity coincide. When is nonlinear, strong monotonicity implies strict monotonicity.
(ii) Whether is linear or nonlinear, we have the following implications [22–24]:
(iii) When and , GUS-property and -property coincide. But, once and are the other cases, for example, and , where denotes the second-order cone, or and , and so forth. GUS-property is not equivalent to -property.
Given an Euclidean Jordan algebra with , from [25, Proposition III 4.4-4.5 and Theorem V.3.7], we know that any Euclidean Jordan algebra and its corresponding symmetric cone are, in a unique way, a direct sum of simple Euclidean Jordan algebras and the constituent symmetric cone therein, respectively, that is, where every is a simple Euclidean Jordan algebra (which cannot be direct sum of two Euclidean Jordan algebras) with the corresponding symmetric cone for , and ( is the dimension of ). Therefore, for any , with , there exist
Through the above description and Cartesian -properties proposed by Chen and Qi  in the setting of semidefinite matrices, Kong et al.  introduced the concept of uniform Cartesian -property for the general transformation in the setting of Euclidean Jordan algebra. This concept is used to study the Lipschitz continuity of the solution mapping in SCCP.
Definition 2.5. Consider a linear or nonlinear transformation . We say that has the uniform Cartesian -property if for any , and , there exist an index and a scalar such that
Remark 2.6. It is easy to observe that when , the uniform Cartesian -property becomes the strong monotonicity of transformation . If and , it becomes the -property in the context of NCP.
When the continuous transformation is linear (i.e., ), we will introduce another concept, the ultra -property of , which is a new concept recently developed for linear transformations on Euclidean Jordan algebra. In fact, the ultra -property is an equivalently straightforward extension of -property in the setting of the semidefinite matrices . Since -property involves the ordinary (associative) product of three square matrices and there may not have an associative (triple) product in an Euclidean Jordan algebra, for this reason, -property cannot be extended in a natural way to an Euclidean Jordan algebra . However, the -property is introduced in Euclidean Jordan algebra using the concepts of principal subtransformation and cone automorphisms of .
Given a Jordan frame in Euclidean Jordan algebra , we define It is known that is a subalgebra of with rank , see [25, Proposition IV.1.1]. By means of Peirce decomposition, we have the following representation : Let denote the orthogonal projection from onto . For a linear transformation , let We call a principal subtransformation of . The determinant of is called a principal minor of .
Definition 2.7 (see ). Consider a linear transformation . We say that has the ultra -property if for any , every principal subtransformation of has the -property.
3. Main Results
In this section, we first give several sufficient conditions for the Lipschitz continuity of the solution mapping in the SCLCP. For the classical LCP and SDLCP, the Lipschitz continuity results have been studied in [9, 13, 14, 19]. Along this direction, we generalize them to general SCCP case where a weaker condition, uniform Cartesian -property, is used. Furthermore, we also establish relationship between the Lipschitz continuity of the solution mapping and the ultra -property.
Theorem 3.1. Let be a continuous linear or nonlinear transformation. If has the uniform Cartesian -property, then is Lipschitz continuous.
Proof. Suppose that has uniform Cartesian -property. From [26, Theorem 6.2], we know that for any , the problem (1.1) has a unique solution, that is, is a single point set. Thus, we let and for any , . If , the inequality is obvious, where . If , from definition of uniform Cartesian -property, there exists an index such that where the third equality follows from because and are the solution of the problem (1.1) for , , respectively. The second inequality is due to , , , and . This implies that . Letting gives . Hence, is Lipschitzian.
Remark 3.2. In Theorem 3.1, if the transformation is linear, the condition of uniform Cartesian -property reduces to the Cartesian -property . However, if we weaken the condition of uniform Cartesian -property to the monotonicity for the linear transformation , the conclusion of Theorem 3.1 is not true. The following example shows that the monotonicity property is not sufficient to conclude that the is Lipschitz continuous with respect to .
Example 3.3. Let be defined as It is obvious that has the monotonicity property. It can be seen that SOL, where is a second-order cone, and is identity element in Euclidean Jordan algebra . Moreover, it is easy to verify that It is an unbounded solution set. However, if the solution mapping of SCLCP is Lipschitz continuous, then SOL must be a bounded set, which is clearly a contradiction.
Kong et al.  proved that the strong monotonicity implies the uniform Cartesian -property whether the transformation is linear or nonlinear. Moreover, when is linear transformation, by [21, Theorem 21], if is self-adjoint and has -property, then is strongly monotone. Hence, we have the following corollary.
Corollary 3.4. Consider Euclidean Jordan algebra .(a)Let be a nonlinear transformation. If is strongly monotone, then is Lipschitz continuous.(b)Let be a linear transformation. If is either(i)strictly monotone, or (ii)self-adjoint and has -property, or (iii)-property and is polyhedral, then is Lipschitz continuous.
Remark 3.5. Even the transformation is linear, the condition of uniform Cartesian -property in Theorem 3.1 or strong monotonicity in Corollary 3.4 cannot be weakened to the GUS-property, otherwise the conclusion is not true. Example 4.2 will illustrate this point.
In the following theorem, we prove that if is Lipschitz continuous, then has the ultra -property provided the linear transformation is monotone. To establish another main result of this paper, the following lemmas play important roles.
Lemma 3.6. Suppose that is Lipschitz continuous, and for some . Then, for all .
If and if has -property i.e., , then has -property.
If is Lipschitz continuous and has -property, then for the every principal subtransformation of , is the Lipschitz continuous with respect to any Jordan frame of .
Lemma 3.7. If is Lipschitz continuous and has -property, then(a)the linear transformation is invertible;(b) for some .
Theorem 3.8. Let be a linear transformation. Suppose is monotone and the solution mapping of is Lipschitz continuous. Then,(a) has the ultra -property;(b) has the GUS-property.
Proof. (a) Consider any Jordan frame of Euclidean Jordan algebra and the principal subtransformation , where for any . Note that
Since is monotone, it follows that the linear transformation and are both monotone. Thus, we have and , where and denote the symmetric cone and the identity element in , respectively. Furthermore, by direct calculation, it is not hard to prove that the solution mapping of SCLCP is Lipschitz continuous if and only if the solution mapping of the corresponding SCLCP is Lipschitz continuous for the linear transformation . Applying Lemma 3.6(a) and (b) yields that has -property. Then using Lemma 3.6(c), we obtain that the solution mapping of the corresponding SCLCP is Lipschitz continuous for the linear transformation . It follows from SOL and Lemma 3.6(a) again that has -property. This together with Lemma 3.7 says that the transformation is invertible.
Next, we want to prove that the transformation has -property. Suppose that an element operator commute with and . Since is monotone by the above analysis, we have which means that . Together with Property 1, it is easy to verify that , and and have the same Jordan frame. Since , we write where is a Jordan frame in , and . Let denote the projection operator from onto the eigenspace of . Then, Let be the principal subtransformation of corresponding to . From the definition of , it follows that . By the same arguments as above, we know that has -property, and the solution mapping of the corresponding SCLCP is Lipschitz continuous for the transformation . Hence, from Lemma 3.7, we get that is invertible. This together with yields , which gives a contradiction to . Therefore, we have proved that has the ultra -property.
(b) This is immediate by [23, Theorem 6.2].
It was shown in [19, Theorem 2.2] that if is monotone and is Lipschitz continuous, then has the -property. Note that -property in is equivalent to the ultra -property in (see ). Therefore, the result of Theorem 3.8 is a natural extension of [19, Theorem 2.2] to the setting of Euclidean Jordan algebra.
4. A Special Linear Transformation
In this section, we specialize to a special linear transformation which is studied in the SCLCP setting, see [19, 23]. For , we consider the corresponding Lyapunov transformation . We will give several equivalent conditions regarding the ultra -property of Lyapunov transformation .
Theorem 4.1. For the Lyapunov transformation , the following statements are equivalent:(a);(b) is strongly monotone;(c) has (uniform) Cartesian -property;(d) has GUS-property;(e) has -property;(f) has the ultra -property;(g) has -property and the solution mapping of the is Lipschitz continuous with respect to .
Proof. (a)(b) For any , we have . Since and , (see [25, Proposition I.1.4]). Thus, has the strong monotonicity property.
(b)(c) It is straightforward by the definitions.
The implication (c)(d) follows from [26, Theorem 6.2].
(d)(e) This follows from [21, Theorem 14].
(e)(a) Suppose that the Lyapunov transformation has -property. Let and , where is a Jordan frame of . Note that if and only if . Suppose that . Let . Then, and operator commute, and . Therefore, by the -property of , we have which leads to .
(b)(f) It follows from [23, Theorem 6.1].
(f)(e) It is obvious.
(b)(h) For any linear transformation, the strong monotonicity is equivalent to the strict monotonicity. Then, it follows from Corollary 3.4 that the solution mapping of the SCLCP is Lipschitz continuous with respect to . Moreover, it is true that the strong monotonicity implies -property for any linear transformation, see . Hence, the conclusion of is obtained.
(h)(b) Suppose that the solution mapping of the SCLCP is Lipschitz continuous with respect to , and has -property. Let be a Jordan frame of and . Note that Since has the -property and the solution map is Lipschitz continuous, (see [27, Theorem 3.1]). It follows from (4.1) that for all . Therefore, the linear transformation has the strong monotonicity. The proof is complete.
In general, the above result may fail to hold. The following example shows that is not Lipschitz continuous, but has the GUS-property. Meanwhile, this example also shows that for Theorem 3.1 and Corollary 3.4, if weaken the condition of strong monotonicity to GUS-property, the conclusions of Theorem 3.1 and Corollary 3.4 are not true.
Example 4.2. Let and . For consider the corresponding Lyapunov transformation defined by It is easy to prove that is positive stable and positive semidefinite, and is a linear transformation. From [10, Theorem 9], we have that has GUS-property. On the other hand, since is not a positive definite matrix, it follows from [19, Theorem 3.3] that is not Lipschitz continuous.
5. Concluding Remarks
In this paper, we have studied the Lipschitz continuity of the solution mapping for symmetric cone linear or nonlinear complementarity problems over Euclidean Jordan algebras and provided several sufficient conditions for the Lipschitz continuity of the solution mapping. We have established the relationship between the Lipschitz continuity of the solution mapping and ultra -property. Furthermore, for Lyapunov transformation, we have shown that the strong monotonicity property, the ultra -property, GUS-property, the Lipschitz continuity of the solution mapping, and so forth are all equivalent to each other.
The authors are grateful to the referees for their constructive comments, which help to improve the paper a lot. The author’s work is supported by National Young Natural Science Foundation (No. 11101302) and The Seed Foundation of Tianjin University (No. 60302041). The author’s work is supported by National Science Council of Taiwan.
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