Volume 2012, Article ID 130682,14pages doi:10.1155/2012/130682
Research Article
Lipschitz Continuity of the Solution Mapping of Symmetric Cone Complementarity Problems
Xin-He Miao
1and Jein-Shan Chen
2, 31Department 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
Correspondence should be addressed to Jein-Shan Chen,[email protected] Received 24 February 2012; Accepted 25 August 2012
Academic Editor: Malisa R. Zizovic
Copyrightq 2012 X.-H. Miao and J.-S. 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.
1. Introduction
LetV, ◦, ·, · we use V for short in subsequent content be a Euclidean Jordan algebra and K be the symmetric cone in V. Given a continuous transformation F : V → V and q ∈ V, the symmetric cone complementarity problem denoted by SCCPF, K, q is to find a vector x ∈ V such that
x ∈ K, Fx q ∈ K,
x, Fx q
0. 1.1
WhenF reduces to a linear transformation L, the above problem is called the symmetric cone linear complementarity problem and is denoted by SCLCPL, K, q, that is, the symmetric cone linear complementarity problem is to find a vectorx ∈ V such that
x ∈ K, Lx q ∈ K,
x, Lx q
0. 1.2
These classes of symmetric cone complementarity problems provide a unified framework for the linear or nonlinear complementarity problemsLCP or NCP, resp. over the nonnegative orthant cone inRn, that is, V Rn andK Rn see 1–4, the second-order cone linear or nonlinear complementarity problems SOCLCP or SOCCP, resp., that is, V Rn and K Kn see 5–8, and the semidefinite linear or nonlinear complementarity problems
SDLCP or SDCP, resp., that is, V Sn andK Sn 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, see3,9.
One of the important issues in complementarity problems is to characterize the Lipschitz continuity of its solutionsor called the Lipschitz continuity of solution mapping
with respect toq. For q ∈ V, let φFq be the set of all solutions to SCCPF, K, q. Then, we intend to know under what conditions the multivalued solution mappingφF : q → φFq
of SCCPF, K, q is Lipschitz continuous. In other words, under what conditions, there will existκ > 0 such that
φF q1
⊆ φF q2
κq1− q2B 1.3
for allq1,q2∈ V satisfying φFq1 / ∅ and φFq2 / ∅, where B is the closed unit ball in V. That is, ifx1∈ φFq1 there exists x2∈ φFq2 such that
x1− x2 ≤ κq1− q2. 1.4
Note that the Lipschitz constantκ depends only on the continuous transformation F. Below is a brief history regarding this issue. For LCPM, q, it is well known that the Lipschitz continuity of the solution mapping with respect toq ∈ V can be described in any one of the following ways:
i the matrix M is P-matrix see 13,14;
ii LCPM, q has a unique solution for all q ∈ Rni.e., GUS-property of M;
iii for any q ∈ Rn, the solution setφMq / ∅ and the set-valued mapping q → φMq
are Lipschitzian.
In particular, Mangasarian and Shiau14 showed that if M is a P-matrix, then solutions of linear inequalities, programs, and LCP are Lipschitz continuous. Murthy et al.15 showed that M is a P-matrix if and only if the LCPM, q has a solution for all q ∈ Rn and the solution mapping is Lipschitzian. Gowda and Sznajder 16 generalized the above result to affine variational inequality problems, while Yen 17 studied Lipschitz continuity of the solution mapping of variational inequalities with a parametric polyhedral constraint. As for NCP, Levy 18 obtained that the solution mapping is locally single-valued and Lipschitz continuous under suitable conditions. How about when K is nonpolyhedral? Balaji et al.
19 proved that L being monotone and the Lipschitz continuity of the solution mapping of SDLCP imply the GUS-property, while Chen and Qi in9 employed Cartesian P-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 20 established the result that if L has the Lipschitzian Q- property, thenL has the positive principal minor property. Gowda et al. 21 showed that if L has P-property, then SCLCPL, K, q has a nonempty compact set for all q ∈ V.
In addition, Tao and Gowda 22 used degree-theoretic arguments to show that under a certain R0-type condition, every P0 symmetric cone nonlinear complementarity problem SCCPF, K, q has a solution. However, it still remains open under what conditions the solution map φF : q → φFq of SCCPF, K, q is Lipschitz continuous. In this paper, we explore new results regarding Lipschitz continuity of the solution mapping of the SCLCPL, K, q or SCCPF, K, q over Euclidean Jordan algebras. In Theorem3.1, we show that if the transformationF has the uniform Cartesian P-property with modulus ρ > 0, then the solution mapping φF is Lipschitz continuous with respect to q ∈ V. Meanwhile, we give examples to show that the solution mapping of nonstrong monotone SCLCPL, K, q
is not Lipschitz continuous with respect to q, and GUS-property does not imply Lipschitz continuity of the solution mapping.
On the other hand, variousP-properties and GUS-property have been investigated in the literature4,9,10,13,16,19,21–24. Relations among them are well studied as well. In 19, Theorem 2.2, it is proved that if the linear transformation L in SDLCP has the monotonicity property and φL is Lipschitzian, then L has the P2-property and the GUS-property. The concept ofP2-property inSnwas extended to a general Euclidean Jordan algebra, called ultra P-property 23. Hence, it is desirable to know whether 19, Theorem 2.2 can be true or not in SCLCPL, K, q if P2-property is replaced by ultraP-property. In this paper, we answer this question positively, see Theorem3.8. Further, for the Lyapunov transformationLa, we present several equivalent conditions for the ultraP-property of La.
Next are a few words about notations and some basic concepts employed. For a
vector
x, x, where ·, · denotes the Euclidean inner product. For the Euclidean Jordan algebraV, let LV denote the set of all continuous linear transformation L : V → V, and AutK denote the set of all invertible linear transformationsΓ : V → V such that ΓK K. For the convex set K, let intK denote the interior of theK. LTmeans the adjoint operator ofL. The identical transformation on V will be denoted byI. For the SCCPF, K, q, the solution set of SCCPF, K, q is denoted by φFq. For the SCLCPL, K, q, the solution set of SCLCPL, K, q is denoted by SOLL, K, q or φLq.
2. Preliminaries
In this section, we briefly recall some basic concepts and background materials in Euclidean Jordan algebras, which will be used in the subsequent analysis. More details can be found in
21–23,25.
An Euclidean Jordan algebra is a tripleV, ◦, ·, · V for short, where V is a finite- dimensional inner product overR and x, y → x ◦ y : V × V → V is a bilinear mapping satisfying the following three conditions:
i x ◦ y y ◦ x for all x, y ∈ V;
ii x ◦ x2◦ y x2◦ x ◦ y for all x, y ∈ V, where x2 x ◦ x;
iii x ◦ y, z x, y ◦ z for all x, y, z ∈ V.
We callx ◦ y the Jordan product of x and y. In addition, if there is an element e ∈ V such that x ◦ e x for all x ∈ V, the element e is called the identity element in V. In a given Euclidean
Jordan algebraV, the set of squares K : {x2:x ∈ V} is a symmetric cone 25, Theorem III.2.1.
In other words,K is a self-dual closed convex cone, and, for any two elements x, y ∈ intK, there exists an invertible linear transformationΓ : V → V such that Γx y and ΓK K.
For anyx ∈ V, we write
x ∈ K x ∈ intK ⇐⇒ x 0 x 0. 2.1
An elementc ∈ V such that c2 c is called an idempotent in V; 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 {e1, e2, . . . , er} of primitive idempotents in V is a Jordan frame if
ei◦ ej 0 for i / j, r
i1
ei e, 2.2
wherer is called the rank of V. Now, we recall the spectral and Peirce decompositions of an elementx in V.
Theorem 2.1 spectral decomposition 25, Theorem III.1.2. Let V be an Euclidean Jordan algebra. Then, there is a numberr such that for every x ∈ V, there exists a Jordan frame {e1, e2, . . . , er} and real numbersλ1, λ2, . . . , λrwith
x λ1e1 · · · λrer. 2.3
Here, the numbersλifori 1, . . . , r are the eigenvalues of x and the expression λ1e1 · · · λrer is the spectral decomposition (or the spectral expansion) ofx.
In an Euclidean Jordan algebraV, corresponding to the convex cone K, let ΠKdenote the metric projection onto K, namely, for an x ∈ V, x∗ ΠKx if and only if x∗ ∈ K and
x−x∗ ∗is unique. For anyx ∈ V, combining the
spectral decomposition ofx with the metric projection of x onto K, we have the expression of metric projectionΠKx as follows see 21:
ΠKx max{0, λ1}e1 · · · max{0, λr}er. 2.4
The Peirce Decomposition
Fix a Jordan frame{e1, e2, . . . , er} in an Euclidean Jordan algebra V. For i, j ∈ {1, 2, . . . , r}, we define the following eigenspaces:
Vii: {x ∈ V | x ◦ ei x} Rei, Vij:
x ∈ V | x ◦ ei 1
2x x ◦ ej
fori / j. 2.5
Theorem 2.2 see 25, Theorem IV.2.1. The space V is the orthogonal direct sum of spaces Vij i ≤ j. Furthermore,
Vij◦ Vij ⊂ Vii Vjj, Vij◦ Vjk⊂ Vik, if i / k, Vij◦ Vkl {0}, if
i, j
∩ {k, l} ∅.
2.6
Hence, given any Jordan frame{e1, e2, . . . , er}, we can write any element x ∈ V as
x r
i1
xiei
i<j
xij, 2.7
wherexi ∈ R and xij ∈ Vij. The expression r
i1xiei
i<jxijis called the Peirce decomposition of x.
Next, we recall concept of Lyapunov transformation and its relevant conclusions which will be used in our analysis later. In an Euclidean Jordan algebraV, for any x ∈ V, we define the corresponding Lyapunov transformationLx : V → V by Lxz x ◦ z for any z ∈ V. As remarked in 21, page 209, traditionally, the notation Lx has been used the Lyapunov transformation 25. As employed in 21, we also reserve the notation Lx
for the Lyapunov transformation and writeLx to denote the image of an element x ∈ V under a linear transformationL : V → V. We say that elements x and y operator commute if LxLy LyLx. It is well known thatx and y operator commute if and only if x and y have their spectral decompositions with respect to a common Jordan frame25, Lemma X.2.2.
Property 1see 21, Proposition 6. For x, y ∈ V, the following conditions are equivalent:
a x 0, y 0, and x, y 0;
b x 0, y 0, and x ◦ y 0.
Moreover, in this case, elements x and y operator commute. That is, x and y have their spectral decompositions with respect to a common Jordan frame.
In fact, from Property1 and definition of1.1, it can be seen that SCCPF, K, q is equivalent to find ax ∈ V such that
x ∈ K, Fx q ∈ K, x ◦
Fx q
0. 2.8
In addition, ifx is a solution of SCCPF, K, q, then x and Fx q operator commute. Now, we review various monotonicity andP-property for a continuous transformation F : V → V.
Definition 2.3. LetV be an Euclidean Jordan algebra. A continuous transformation F : V → V is said to be
a monotone if Fx − Fy, x − y ≥ 0, for all x, y ∈ V;
b strictly monotone if Fx − Fy, x − y > 0, for all x / y ∈ V;
c strongly monotone if there is α > 0 such that
Fx − F y
, x − y≥ αx − y2, ∀x, y ∈ V. 2.9
It is said to have
d GUS-property if SCCPF, K, q has a unique solution for any q ∈ V;
e P-property if
x − y and Fx − F y
operator commute
x − y
◦
Fx − F y
0
⇒ x y; 2.10
f Q-property if φFq / ∅ for any q ∈ V.
Remark 2.4. i When F is linear, strict monotonicity and strong monotonicity coincide. When F is nonlinear, strong monotonicity implies strict monotonicity.
ii Whether F is linear or nonlinear, we have the following implications 22–24:
strong monotonicity⇒ strict monotonicity ⇒ P-property ⇒ Q-property,
strong monotonicity⇒ GUS-property ⇒ P-property. 2.11
iii When V Rn andK Rn, GUS-property andP-property coincide. But, once V andK are the other cases, for example, V RnandK Kn, whereKndenotes the second- order cone, orV SnandK Sn, and so forth. GUS-property is not equivalent toP-property.
Given an Euclidean Jordan algebraV with dimV n > 1, from 25, Proposition III 4.4-4.5 and Theorem V.3.7, we know that any Euclidean Jordan algebra V and its corresponding symmetric cone K are, in a unique way, a direct sum of simple Euclidean Jordan algebras and the constituent symmetric cone therein, respectively, that is,
V V1× · · · × Vm, K K1× · · · × Km, 2.12
where everyVi is a simple Euclidean Jordan algebrawhich cannot be direct sum of two Euclidean Jordan algebras with the corresponding symmetric cone Kifori 1, . . . , m, and n m
i1niniis the dimension ofVi. Therefore, for any x x1, . . . , xmT,y y1, . . . , ymT ∈ V with xi, yi∈ Vi, there exist
x ◦ y
x1◦ y1, . . . , xm◦ ymT
∈ V, x, y
x1, y1
· · · xm, ym
. 2.13
Through the above description and CartesianP-properties proposed by Chen and Qi
9 in the setting of semidefinite matrices, Kong et al. 26 introduced the concept of uniform CartesianP-property for the general transformation F in the setting of Euclidean Jordan alge- bra. This concept is used to study the Lipschitz continuity of the solution mapping in SCCP.
Definition 2.5. Consider a linear or nonlinear transformationF : V → V. We say that F has the uniform CartesianP-property if for any x, y ∈ V and x / y, there exist an index ν ∈ {1, 2, . . . , m}
and a scalarρ > 0 such that
x − y
ν,
Fx − F y
ν
≥ ρx − y2. 2.14
Remark 2.6. It is easy to observe that whenm 1, the uniform Cartesian P-property becomes the strong monotonicity of transformationF. If m n and V Rn, it becomes theP-property in the context of NCP.
When the continuous transformation F : V → V is linear i.e., F L, we will introduce another concept, the ultra P-property of L, which is a new concept recently developed for linear transformations on Euclidean Jordan algebra. In fact, the ultra P- property is an equivalently straightforward extension of P2-property in the setting of the semidefinite matrices23. Since P2-property involves the ordinaryassociative product of three square matrices and there may not have an associativetriple product in an Euclidean Jordan algebra, for this reason, P2-property cannot be extended in a natural way to an Euclidean Jordan algebra23. However, the P2-property is introduced in Euclidean Jordan algebra using the concepts of principal subtransformation and cone automorphisms ofV 23.
Given a Jordan frame{e1, e2, . . . , er} in Euclidean Jordan algebra V, we define
Vl Ve1 · · · el, 1 : {x ∈ V | x ◦ e1 · · · el x} for 1 ≤ l ≤ r. 2.15
It is known thatVlis a subalgebra ofV with rank l, see 25, Proposition IV.1.1. By means of Peirce decomposition, we have the following representation21:
Vl Re1 · · · Rel
i<j≤l
Vij. 2.16
LetPldenote the orthogonal projection from V onto Vl. For a linear transformation L : V → V, let
Ll L{e1,...,el}: PlL : Vl−→ Vl. 2.17
We callLla principal subtransformation ofL. The determinant of Llis called a principal minor ofL.
Definition 2.7see 23. Consider a linear transformation L : V → V. We say that L has the ultraP-property if for any Γ ∈ AutK, every principal subtransformation of L ΓTLΓ has theP-property.
3. Main Results
In this section, we first give several sufficient conditions for the Lipschitz continuity of the solution mappingφLin the SCLCPL, K, q. 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 SCCPF, K, q case where a weaker condition, uniform Cartesian P-property, is used. Furthermore, we also establish relationship between the Lipschitz continuity of the solution mapping and the ultraP-property.
Theorem 3.1. Let F : V → V be a continuous linear or nonlinear transformation. If F has the uniform CartesianP-property, then φFis Lipschitz continuous.
Proof. Suppose thatF has uniform Cartesian P-property. From 26, Theorem 6.2, we know that for anyq ∈ V, the problem 1.1 has a unique solution, that is, φFq is a single point set. Thus, we let{x} φFq1 and {y} φFq2 for any q1,q2 ∈ V. If x y, the inequality
1− q2 is obvious, where κ > 0. If x / y, from definition of uniform Cartesian P-property, there exists an index ν ∈ {1, . . . , m} such that
ρx − y2≤
x − y
ν,
Fx − F y
ν
xν− yν, Fxν− F y
ν
xν− yν,
Fx q1
ν− F
y
q2
ν
−
xν− yν, q1
ν− q2
ν
xν− yν, q2
ν− q1
ν
− xν,
F y
q2
ν
− yν,
Fx q1
ν
≤
xν− yν, q2
ν− q1
ν
≤xν− yνq1
ν− q2
ν
≤x − yq1− q2,
3.1
where the third equality follows fromxν, Fx q1ν 0 yν, Fy q2ν because x andy are the solution of the problem 1.1 for q1,q2 ∈ V, respectively. The second inequality is due toxν,yν,Fx q1ν, andFy q2ν ∈ Kν. This implies that 1− q2. Letting 1− q2. Hence, φFis Lipschitzian.
Remark 3.2. In Theorem 3.1, if the transformation F is linear, the condition of uniform CartesianP-property reduces to the Cartesian P-property 26. However, if we weaken the condition of uniform CartesianP-property to the monotonicity for the linear transformation L, the conclusion of Theorem 3.1 is not true. The following example shows that the monotonicity property is not sufficient to conclude that the φLis Lipschitz continuous with respect toq ∈ V.
Example 3.3. LetL : R3 → R3be defined as
L :
⎡
⎣0 0 0 0 1 0 0 0 2
⎤
⎦, where L
⎛
⎝
⎡
⎣x y z
⎤
⎦
⎞
⎠ :
⎡
⎣0 y 2z
⎤
⎦. 3.2
It is obvious thatL has the monotonicity property. It can be seen that SOLL, K3, e {0}, whereK3 ⊂ R3is a second-order cone, ande is identity element in Euclidean Jordan algebra R3. Moreover, it is easy to verify that
α, 0, 0T : 0< α ∈ R
⊆ SOL
L, K3, 0
. 3.3
It is an unbounded solution set. However, if the solution mapping φL of SCLCPL, K3, 0
is Lipschitz continuous, then SOLL, K3, 0 must be a bounded set, which is clearly a contradiction.
Kong et al.26 proved that the strong monotonicity implies the uniform Cartesian P- property whether the transformationF is linear or nonlinear. Moreover, when F L is linear transformation, by21, Theorem 21, if L is self-adjoint and has P-property, then L is strongly monotone. Hence, we have the following corollary.
Corollary 3.4. Consider Euclidean Jordan algebra V.
a Let F : V → V be a nonlinear transformation. If F is strongly monotone, then φF is Lipschitz continuous.
b Let L : V → V be a linear transformation. If L is either
i strictly monotone, or
ii self-adjoint and has P-property, or
iii P-property and K is polyhedral, thenφLis Lipschitz continuous.
Remark 3.5. Even the transformation F is linear, the condition of uniform Cartesian P- property in Theorem3.1or strong monotonicity in Corollary3.4cannot be weakened to the GUS-property, otherwise the conclusion is not true. Example4.2will illustrate this point.
In the following theorem, we prove that ifφLis Lipschitz continuous, thenL has the ultraP-property provided the linear transformation L is monotone. To establish another main result of this paper, the following lemmas play important roles.
Lemma 3.6. a Suppose that φLis Lipschitz continuous, and SOLL, K, q {0} for some q 0.
Then, SOLL, K, q {0} for all q 0.
b If SOLL, K, e {0} and if L has R0-propertyi.e., SOLL, K, 0 {0}, then L has Q-property.
c If φLis Lipschitz continuous andL has Q-property, then for the every principal sub- transformationLlofL, φLl is the Lipschitz continuous with respect to any Jordan frame ofV.
Proof. Please see20, Lemma 5 for part a, 20, Proposition 3 for part b, and 20, Lemma 4 for part c.
Lemma 3.7. If φLis Lipschitz continuous andL has Q-property, then
a the linear transformation L is invertible;
b SOLL, K, q {0} for some q 0.
Proof. Parta is from 20, Lemma 6, while part b is from 20, Lemma 1.
Theorem 3.8. Let L : V → V be a linear transformation. Suppose L is monotone and the solution mappingφLof SCLCPL, K, q is Lipschitz continuous. Then,
a L has the ultra P-property;
b L has the GUS-property.
Proof. a Consider any Jordan frame {e1, . . . , er} of Euclidean Jordan algebra V and the prin- cipal subtransformation Ll : L{e1,...,el} : Vl → Vl, where L ΓTLΓ for any Γ ∈ AutK.
Note that
z, Lz
z, ΓTLΓz
Γz, LΓz for any z ∈ V,
z1, Llz1
z1,
PlL
z1
z1, Lz1
, where z1∈ Vl⊆ V.
3.4
SinceL is monotone, it follows that the linear transformation L and Ll are both monotone.
Thus, we have SOLL, K, e {0} and SOLLl, Kl, el {0}, where Kland el denote the symmetric cone and the identity element in Vl, respectively. Furthermore, by direct calculation, it is not hard to prove that the solution mappingφLof SCLCPL, K, q is Lipschitz continuous if and only if the solution mappingφLof the corresponding SCLCP is Lipschitz continuous for the linear transformation L. Applying Lemma 3.6a and b yields that L has Q-property. Then using Lemma3.6c, we obtain that the solution mapping φLl of the corresponding SCLCP is Lipschitz continuous for the linear transformation Ll. It follows from SOLLl, Kl, el {0} and Lemma3.6a again that LlhasQ-property. This together with Lemma3.7says that the transformation Llis invertible.
Next, we want to prove that the transformation Ll has P-property. Suppose that an element 0/ x ∈ Vloperator commute with Llx and x ◦ Llx 0. Since Llis monotone by the above analysis, we have
0≤
x, Llx
x ◦ Llx, el
≤ 0, 3.5
which means thatLlx ◦ x, el 0. Together with Property 1, it is easy to verify that
Llx ◦ x 0, and Llx and x have the same Jordan frame. Since Llx ◦ x 0, we write
x k
i1
λifi, Llx l
ik1
μifi, 3.6
where {f1, f2, . . . , fl} is a Jordan frame in Vl,λi/ 0 i 1, . . . , k and 1 ≤ k ≤ l. Let Qk
denote the projection operator fromVlonto the eigenspaceWkoff1 · · · fk. Then,
0 QkLlx QkL{e1,...,el}x QkPlLx. 3.7
Let Tk : QkLl : Wk → Wk be the principal subtransformation of Ll corresponding to{1, . . . , k}. From the definition of Tk, it follows that Tkx QkLlx 0. By the same
arguments as above, we know that Tk has Q-property, and the solution mapping φTk of the corresponding SCLCP is Lipschitz continuous for the transformation Tk. Hence, from Lemma3.7, we get thatTkis invertible. This together withTkx 0 yields x 0, which gives a contradiction tox / 0. Therefore, we have proved that L has the ultra P-property.
b This is immediate by 23, Theorem 6.2.
It was shown in19, Theorem 2.2 that if L : Sn → Snis monotone andφLis Lipschitz continuous, thenL has the P2-property. Note thatP2-property inSnis equivalent to the ultra P-property in Snsee 23. Therefore, the result of Theorem3.8is a natural extension of19, 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, see19, 23. For a ∈ V, we consider the corresponding Lyapunov transformation La. We will give several equivalent conditions regarding the ultraP-property of Lyapunov transformationLa.
Theorem 4.1. For the Lyapunov transformation La a ∈ V, the following statements are equivalent:
a a 0;
b Lais strongly monotone;
c Lahas (uniform) CartesianP-property;
d Lahas GUS-property;
e LahasP-property;
f Lahas the ultraP-property;
g La has Q-property and the solution mapping φLa of the SCLCPLa, K, q is Lipschitz continuous with respect toq ∈ V.
Proof. a⇒b For any 0 / x ∈ V , we have x, Lax x, a ◦ x a, x2. Since a 0 andx2 ∈ K, a, x2 > 0 see 25, Proposition I.1.4. Thus, Lahas the strong monotonicity property.
b⇒c It is straightforward by the definitions.
The implicationc⇒d follows from 26, Theorem 6.2.
d⇒e This follows from 21, Theorem 14.
e⇒a Suppose that the Lyapunov transformation La has P-property. Let a r
i1λiaei and I {i : λia ≤ 0}, where {e1, . . . , er} is a Jordan frame of V. Note that a 0 if and only if I ∅. Suppose that I / ∅. Let x
i∈Iei/ 0. Then, x and Lax operator commute, andx ◦ Lax
i∈Iλiaei 0. Therefore, by the P-property of La, we havex 0 which leads toa 0.
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 Corollary3.4that the solution mappingφLa of the SCLCPLa, K, q is Lipschitz continuous with respect to q ∈ V. Moreover, it is true that the strong monotonicity impliesQ-property for any linear transformation, see 21. Hence, the conclusion ofh is obtained.
h⇒b Suppose that the solution mapping φLa of the SCLCPLa, K, q is Lipschitz continuous with respect toq ∈ V, and LahasQ-property. Let {e1, . . . , er} be a Jordan frame of V and x r
i1λixei. Note that
x, Lax r
i,j1
λixλjx ei, La
ej
r
i1
λ2ixei, Laei. 4.1
SinceLahas theQ-property and the solution map φLais Lipschitz continuous,Laei, ei > 0
see 27, Theorem 3.1. It follows from 4.1 that Lax, x > 0 for all 0 / x ∈ V. Therefore, the linear transformationLahas the strong monotonicity. The proof is complete.
In general, the above result may fail to hold. The following example shows thatφLis not Lipschitz continuous, butL has the GUS-property. Meanwhile, this example also shows that for Theorem3.1and Corollary3.4, if weaken the condition of strong monotonicity to GUS-property, the conclusions of Theorem3.1and Corollary3.4are not true.
Example 4.2. LetV S2andK S2. For
A :
0 −3 3 3
, 4.2
consider the corresponding Lyapunov transformation defined by
LAX : AX XAT. 4.3
It is easy to prove that A is positive stable and positive semidefinite, and LA is a linear transformation. From 10, Theorem 9, we have that LA has GUS-property. On the other hand, since A is not a positive definite matrix, it follows from 19, Theorem 3.3 that φLA 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 ultraP-property. Furthermore, for Lyapunov transformation, we have shown that the strong monotonicity property, the ultra P-property, GUS-property, the Lipschitz continuity of the solution mapping, and so forth are all equivalent to each other.
Acknowledgments
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 FoundationNo. 11101302 and The Seed Foundation of Tianjin University No. 60302041.
The author’s work is supported by National Science Council of Taiwan.
References
1 R. W. Cottle, J.-S. Pang, and R. E. Stone, The Linear Complementarity Problem, Computer Science and Scientific Computing, Academic Press, Boston, Mass, USA, 1992.
2 R. W. Cottle, J.-S. Pang, and V. Venkateswaran, “Sufficient matrices and the linear complementarity problem,” Linear Algebra and Its Applications, vol. 114-115, pp. 231–249, 1989.
3 F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems. Vol.
I, Springer Series in Operations Research, Springer, New York, NY, USA, 2003.
4 M. S. Gowda, “On the extended linear complementarity problem,” Mathematical Programming, vol. 72, no. 1, pp. 33–50, 1996.
5 J.-S. Chen, X. Chen, and P. Tseng, “Analysis of nonsmooth vector-valued functions associated with second-order cones,” Mathematical Programming, vol. 101, no. 1, pp. 95–117, 2004.
6 J.-S. Chen and P. Tseng, “An unconstrained smooth minimization reformulation of the second-order cone complementarity problem,” Mathematical Programming, vol. 104, no. 2-3, pp. 293–327, 2005.
7 M. Fukushima, Z. Q. Lou, and P. Tseng, “Smoothing functions for secondordercone complementarity problems,” SIAM Journal on Optimization, vol. 12, no. 2, pp. 436–460, 2002.
8 S. Pan and J.-S. Chen, “A damped Gauss-Newton method for the second-order cone complementarity problem,” Applied Mathematics and Optimization, vol. 59, no. 3, pp. 293–318, 2009.
9 X. Chen and H. Qi, “Cartesian P-property and its applications to the semidefinite linear complementarity problem,” Mathematical Programming, vol. 106, no. 1, pp. 177–201, 2006.
10 M. S. Gowda and Y. Song, “On semidefinite linear complementarity problems,” Mathematical Program- ming, vol. 88, no. 3, pp. 575–587, 2000.
11 D. Sun, “The strong second-order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications,” Mathematics of Operations Research, vol. 31, no. 4, pp. 761–776, 2006.
12 P. Tseng, “Merit functions for semi-definite complementarity problems,” Mathematical Programming, vol. 83, no. 2, pp. 159–185, 1998.
13 M. S. Gowda, “On the continuity of the solution map in linear complementarity problems,” SIAM Journal on Optimization, vol. 2, no. 4, pp. 619–634, 1992.
14 O. L. Mangasarian and T. H. Shiau, “Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems,” SIAM Journal on Control and Optimization, vol. 25, no. 3, pp. 583–595, 1987.
15 G. S. R. Murthy, T. Parthasarathy, and M. Sabatini, “Lipschitzian Q-matrices are P-matrices,” Mathe- matical Programming, vol. 74, no. 1, pp. 55–58, 1996.
16 M. S. Gowda and R. Sznajder, “On the Lipschitzian properties of polyhedral multifunctions,”
Mathematical Programming, vol. 74, no. 3, pp. 267–278, 1996.
17 N. D. Yen, “Lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint,” Mathematics of Operations Research, vol. 20, no. 3, pp. 695–708, 1995.
18 A. B. Levy, “Stability of solutions to parameterized nonlinear complementarity problems,” Mathemat- ical Programming, vol. 85, no. 2, pp. 397–406, 1999.
19 R. Balaji, T. Parthasarathy, D. Sampangi Raman, and V. Vetrivel, “On the Lipschitz continuity of the solution map in semidefinite linear complementarity problems,” Mathematics of Operations Research, vol. 30, no. 2, pp. 462–471, 2005.
20 R. Balaji, “On an interconnection between the Lipschitz continuity of the solution map and the posi- tive principal minor property in linear complementarity problems over Euclidean Jordan algebras,”
Linear Algebra and Its Applications, vol. 426, no. 1, pp. 83–95, 2007.
21 M. S. Gowda, R. Sznajder, and J. Tao, “Some P-properties for linear transformations on Euclidean Jordan algebras,” Linear Algebra and Its Applications, vol. 393, pp. 203–232, 2004.
22 J. Tao and M. S. Gowda, “Some P-properties for nonlinear transformations on Euclidean Jordan algebras,” Mathematics of Operations Research, vol. 30, no. 4, pp. 985–1004, 2005.
23 M. S. Gowda and R. Sznajder, “Automorphism invariance of P- and GUS-properties of linear transfor- mations on Euclidean Jordan algebras,” Mathematics of Operations Research, vol. 31, no. 1, pp. 109–123, 2006.
24 M. S. Gowda and R. Sznajder, “Some global uniqueness and solvability results for linear comple- mentarity problems over symmetric cones,” SIAM Journal on Optimization, vol. 18, no. 2, pp. 461–481, 2007.
25 J. Faraut and A. Kor´anyi, Analysis on Symmetric Cones, Oxford Mathematical Monographs, The Clarendon Press; Oxford University Press, New York, NY, USA, 1994.
26 L. Kong, L. Tunc¸el, and N. Xiu, “Vector-valued implicit Lagrangian for symmetric cone complemen- tarity problems,” Asia-Pacific Journal of Operational Research, vol. 26, no. 2, pp. 199–233, 2009.
27 I. Jeyaraman and V. Vetrivel, “On the Lipschitzian property in linear complementarity problems over symmetric cones,” Linear Algebra and Its Applications, vol. 435, no. 4, pp. 842–851, 2011.
