The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra
aa r X i v : . [ m a t h . O C ] O c t Noname manuscript No. (will be inserted by the editor)
The weighted horizontal linear complementarityproblem on a Euclidean Jordan algebra
Xiaoni Chi · M. Seetharama Gowda · Jiyuan Tao
Received: date / Accepted: date
Abstract
A weighted complementarity problem (wCP) is to find a pair ofvectors belonging to the intersection of a manifold and a cone such that theproduct of the vectors in a certain algebra equals a given weight vector. Ifthe weight vector is zero, we get a complementarity problem. Examples ofsuch problems include the Fisher market equilibrium problem and the linearprogramming and weighted centering problem. In this paper we consider theweighted horizontal linear complementarity problem (wHLCP) in the settingof Euclidean Jordan algebras and establish some existence and uniquenessresults. For a pair of linear transformations on a Euclidean Jordan algebra, weintroduce the concepts of R , R , and P properties and discuss the solvability ofwHCLPs under nonzero (topological) degree conditions. A uniqueness result isstated in the setting of R n . We show how our results naturally lead to interiorpoint systems. Keywords
Weighted horizontal linear complementarity problem · EuclideanJordan algebra · Degree · R -pair Mathematics Subject Classification (2010)
Xiaoni ChiSchool of Mathematics and Computing Science, Guangxi Colleges and Universities KeyLaboratory of Data Analysis and Computation, Guilin University of Electronic Technology,Guilin 541004, Guangxi, P. R. ChinaE-mail: [email protected]. Seetharama Gowda ( (cid:12) )Department of Mathematics and Statistics, University of Maryland, Baltimore County, Bal-timore, Maryland 21250, USAE-mail: [email protected] TaoDepartment of Mathematics and Statistics, Loyola University Maryland, Baltimore, Mary-land 21210, USAE-mail: [email protected] Xiaoni Chi et al.
Introduced in [13], a weighted complementarity problem (wCP) is to find apair of vectors ( x, y ) belonging to the intersection of a manifold with a conesuch that their product in a certain (Euclidean Jordan) algebra equals a givenweight vector w . When w is the zero vector, wCP becomes a complementarityproblem (CP). To elaborate, consider a Euclidean Jordan algebra ( V, ◦ , h· , ·i )with symmetric cone V + [5]. Given a map F : V × V × R l → V × R l and aweight vector w ∈ V + , wCP is to find ( x, y ) ∈ V × V such that for some u , x ∈ V + , y ∈ V + ,x ◦ y = w,F ( x, y, u ) = 0 . (1)Deferring this general problem for a future study, in [13] and [14], Potra studiesaffine wCP on the (Euclidean Jordan) algebra R n with several examples andresults. Given matrices A, B ∈ R ( n + m ) × n , C ∈ R ( n + m ) × m , a weight vector w ∈ R n + , and q ∈ R n + m , the weighted mixed horizontal linear complementarityproblem considered in [13], [14] is to find ( x, y, z ) ∈ R n × R n × R m such that x ≥ , y ≥ ,x ∗ y = w,Ax + By + Cz = q, (2)where x ∗ y denotes the Hadamard (= componentwise) product of vectors x and y . Here, C is assumed to be of full column rank and so, with a suitablechange of variables, one could transform (see Section 2 in [14]) the aboveproblem to an equivalent affine wCP where C becomes vacuous and A and B are square. It is shown in [13] that the Fisher market equilibrium problem[3],[17] and the linear programming and weighted centering problem [1] canbe formulated in the form (2), where the triple ( A, B, C ) satisfies a certainmonotonicity condition. In [13], Potra presented and analyzed two interior-point methods for solving such monotone affine wCPs. Subsequently, replacing‘monotone’ conditions by ‘row and column sufficient’ conditions, Potra [14]described several theoretical results and a corrector-predictor interior-pointmethod for its numerical solution.We note that weighted complementarity problems were studied much ear-lier in connection with interior point methods. For example, in [8], Kojima etal showed that if (continuous) f : R n → R n is a uniform P-function, that is,there exists some γ > ≤ i ≤ n ( x i − y i ) (cid:16) f i ( x ) − f i ( y ) (cid:17) ≥ γ || x − y || (for all x, y ∈ R n + ) , then the mapping b f : R n + × R n + → R n + × R n is a homeomorphism, where b f ( x, y ) = (cid:20) x ∗ yy − f ( x ) (cid:21) . eighted complementarity problem 3 In this situation, the following weighted nonlinear complementarity problemhas a unique solution for each w ∈ R n + and q ∈ R n : x ≥ , y ≥ ,x ∗ y = w,y = f ( x ) + q. (3)Another work that specifically looks at the general wCP (1) is by Yoshise[18]. In this work, under certain‘monotonicity and injectivity’ assumptions, itis shown that the map ( x, y, u ) → (cid:16) x ◦ y, F ( x, y, u ) (cid:17) is a homeomorphism on acertain subset of V + × V + × R l , leading to the solvability of wCP, see Theorem3.10 and Corollary 4.4 in [18].Our objective in this paper is to consider the following affine wCP in thesetting of Euclidean Jordan algebras. Given two linear transformations A and B on a Euclidean Jordan algebra V , a weight vector w ∈ V + , and q ∈ V , the weighted horizontal linear complementarity problem wHLCP( A, B, w, q ) is tofind ( x, y ) ∈ V × V such that x ≥ , y ≥ ,x ◦ y = w,Ax + By = q, (4)where x ≥ x ∈ V + , etc. If w = 0, the above problem reduces tothe (symmetric cone) horizontal linear complementarity problem on V , denotedby HLCP( A, B, q ). If w = 0, A = I , and B = − M , HLCP( A, B, q ) reducesto the (symmetric cone) linear complementarity problem
LCP(
M, q ) on V . Inparticular, when V = R n , this reduces to the standard linear complementarityproblem .Our analysis differs from Potra’s [13], [14] in several ways. First, our settingis that of a general Euclidean Jordan algebra, instead of R n . Second, insteadof the ‘monotone/sufficient’ conditions, we rely on the R property (that iscommonly used in the LCP literature) coupled with a nonzero degree con-dition of a certain map associated with wHLCP. Third, instead of using theoptimization methodology, we rely on the degree theoretic tools. Our analysisalso differs from that of Yoshise [18] where results were proved under certain‘monotonicity and injectivity’ conditions.In this paper, we establish some basic existence/uniqueness results aboutwHLCPs. Generalizing the LCP concept of a degree of an R -matrix, weintroduce the concept of degree of an R -pair of linear transformations in thesetting of Euclidean Jordan algebras. Assuming that this degree is nonzero forthe pair { A, B } , we show that wHLCP( A, B, w, q ) has a nonempty compactsolution set for every ( w, q ) ∈ V + × V. This conclusion, in particular, will allowus to say that • the map Γ : V + × V + → V + × V given by Γ ( x, y ) = (cid:16) x ◦ y, Ax + By (cid:17) issurjective and Xiaoni Chi et al. • when w > w ∈ int( V + )), the ‘interior point system’ x > , y > , x ◦ y = w, and Ax + By = q has a nonempty compact solution set.The result about the interior point system appears to be new even in thesetting of standard LCPs.We also introduce the concept of a P -pair and show that when V = R n ,wHLCP( A, B, w, q ) has a unique solution for every ( w, q ) ∈ R n + × R n .The organization of our paper is as follows. In Section 2, we cover somebasic material. In Section 3, we introduce the concepts of R and R pairs anddefine the degree of an R -pair. Section 4 covers the main result of the paperdescribing the solvability of wHLCP. While Section 5 deals with the solutionset behavior, Sections 6 and 7 cover P -pairs and address uniqueness issues. Throughout this paper, R n denotes the Euclidean n -space of real column vec-tors. We use the (same) symbol 0 to denote the zero vector in any vector space.( V, ◦ , h· , ·i ) denotes a Euclidean Jordan algebra of rank n with symmetric cone V + [5], [7]. Here, x ◦ y and h x, y i , respectively, denote the Jordan product andthe inner product of elements x and y . The unit element of V is denoted by e .For a subset S of V , the interior, closure, and boundary are denoted by int ( S ), S , and ∂ ( S ). If x ∈ V + ( x ∈ int ( V + )), we write x ≥ x > x ∈ V , x + denotes the projection of x onto V + , and we let x − := x + − x , | x | := x + + x − . These can also be described via the spectral decomposition x = P n x i e i (where x , x , . . . , x n are the eigenvalues of x and { e , e , . . . , e n } is Jordan frame): x + = P n x + i e i , | x | = P n | x i | e i , etc. We see that | x | = x , √ x = | x | , h x + , x − i = 0 and x + ◦ x − = 0 . For x, y ∈ V , we define x ⊓ y := x − ( x − y ) + . When V = R n (with the usual componentwise product and the inner product),this reduces to min { x, y } , the componentwise minimum of (vectors) x and y in R n . For this reason, we may call the map ( x, y ) → x ⊓ y , the ‘min map’ on V . The map ( x, y ) → x + y − p x + y is called the Fischer-Burmeister map. It has been extensively used in the com-plementarity literature. Below, we state some basic properties of these twomaps. Proposition 1
The following statements hold in V : ( i ) u + x ⊓ y = ( u + x ) ⊓ ( u + y ) . ( ii ) λ ( x ⊓ y ) = λx ⊓ λy for all λ ≥ . ( iii ) The following are equivalent: eighted complementarity problem 5 ( a ) x ⊓ y = 0 . ( b ) x ≥ , y ≥ , and h x, y i = 0 . ( c ) x ≥ , y ≥ , and x ◦ y = 0 . Moreover, in each case, x and y operator commute.(iv) When w ≥ , the following are equivalent: ( a ) x + y − p x + y + 2 w = 0( b ) x ≥ , y ≥ , and x ◦ y = w. Moreover, when w = 0 or w = e (the unit element of V ), above x and y operator commute. Proof.
Items ( i ) and ( ii ) follow easily from the definition of ‘min map’. Item( iii ) appears in [7], Proposition 6 and Item ( iv ) for w = 0 or w = e is covered in[7], Propositions 6 and 7. Now, let w ≥ x + y − p x + y + 2 w =0. Then, x + y = p x + y + 2 w . This shows that x + y ≥ x ◦ y = w . We need only show that x ≥ y ≥
0. Considerthe spectral expansion x = λ e + λ e + · · · + λ n e n , where λ , λ , . . . , λ n arethe eigenvalues of x and { e , e , . . . , e n } is a Jordan frame in V . Suppose, ifpossible, x
0; we may assume without loss of generality that λ <
0. Then, x ◦ e = λ e and0 ≤ h x + y, e i = h x, e i + h y, e i = λ || e || + 1 λ h y, x ◦ e i = λ || e || + 1 λ h x ◦ y, e i = λ || e || + 1 λ h w, e i < , as h w, e i ≥
0. This contradiction proves that all eigenvalues of x are nonneg-ative; so x ≥
0. Similarly, y ≥
0. Thus we have ( iv ). ⊓⊔ Item ( iv ) in the above proposition will allow us to formulate a wHLCP asa system of equations. In fact, ( x, y ) is a solution of wHLCP( A, B, w, q ) (4) ifand only if it is a solution of the system x + y − p x + y + 2 w = 0 ,Ax + By − q = 0 . Our next key result will be used to show that the min and the Fischer-Burmeister maps are ‘homotopic’. This will allow us to replace the Fischer-Burmeister map by the ‘simpler’ min map in our main solution analysis.
Proposition 2
Let x, y ∈ V and ≤ t ≤ . Then, t h x + y − p x + y i + (1 − t ) x ⊓ y = 0 ⇐⇒ x ⊓ y = 0 . Proof.
In view of Items ( iii ) and ( iv ) in the previous proposition, we proveonly the ‘if’ part. We also assume without loss of generality, 0 < t <
1. Let u := t h x + y − p x + y i . From Item ( i ) of the previous proposition, h (1 − t ) x + u i ⊓ h (1 − t ) y + u i = 0 . Xiaoni Chi et al.
This implies that (1 − t ) x + u ≥ − t ) y + u ≥
0. Now, (1 − t ) x + u ≥ − t ) x + t h x + y − p x + y i ≥
0, that is, x + ty ≥ t p x + y . As p x + y ≥ p y = | y | (which is a consequence of the so-called L¨owner-Heinz inequality, see [7], Proposition 8), we see that x ≥ t | y | − ty ≥ . Hence x ≥
0. Similarly, y ≥
0. It follows that h x, y i ≥
0. We now show that h x, y i ≤ h x, y i = 0.We first note that x ⊓ y = ( x + y − | x − y | ) /
2. Let p := t − t h x + y − p x + y i so that x + y − p x + y = αp , where α := − tt . Then, p x + y = ( x + y ) − α p. Squaring both sides and simplifying, we get p ◦ ( x + y ) = 12 α h x ◦ y + α p i . (5)As t h x + y − p x + y i + (1 − t ) x ⊓ y = 0, we have p + x ⊓ y = 0 , that is,2 p + ( x + y ) = | x − y | . Squaring both sides, noting | x − y | = ( x − y ) , and simplifying, we get4 p + 2 x ◦ y + 4 p ◦ ( x + y ) = − x ◦ y. We replace 4 p ◦ ( x + y ) by using (5) to get an expression of the form β x ◦ y + γ p = 0 , where numbers β and γ are positive. This yields x ◦ y ≤ h x, y i = h x ◦ y, e i ≤ . Finally, since h x, y i ≥ , we have h x, y i = 0 . Thus we have shownthat x, y ≥ h x, y i = 0 . Hence, x ⊓ y = 0 . ⊓⊔ We end this subsection by quoting a well-known determinantal formula.
Proposition 3 [12] For
A, B, X, Y ∈ R n × n , with X, Y commuting, the fol-lowing formula holds: det (cid:20) A − BX Y (cid:21) = det( AY + BX ) . A similar statement can be made about linear transformations. eighted complementarity problem 7 Ω is a boundedopen set in R n , g : Ω → R n is continuous and p g ( ∂Ω ), where Ω and ∂Ω denote, respectively, the closure and boundary of Ω. Then the degree of g over Ω with respect to p is defined; it is an integer and will be denoted bydeg( g, Ω, p ). When this degree is nonzero, the equation g ( x ) = p has a solutionin Ω. Suppose g ( x ) = p has a unique solution, say, x ∗ in Ω. Then deg( g, Ω ′ , p ),which equals deg( g, Ω ′ , g ( x ∗ )), is constant over all bounded open sets Ω ′ con-taining x ∗ and contained in Ω. This common degree is called the (topological)index of g at x ∗ ; it will be denoted by ind( g, x ∗ ). In particular, if g : R n → R n is a continuous map such that g ( x ) = 0 ⇔ x = 0, then for any bounded openset containing 0, we have ind( g,
0) = deg( g, Ω, g is the identity map, ind( g,
0) = 1 . Let H ( x, t ) : R n × [0 , → R n be continuous (in which case, we say that H isa homotopy). Suppose that for some bounded open set Ω in R n , H ( ∂Ω, t )for all t ∈ [0 , . Then, the homotopy invariance property of degree says thatdeg (cid:16) H ( · , t ) , Ω, (cid:17) is independent of t. In particular, if the zero set n x : H ( x, t ) = 0 for some t ∈ [0 , o is bounded, then for any bounded open set Ω in R n containing this zero set,we have deg (cid:16) H ( · , , Ω, (cid:17) = deg (cid:16) H ( · , , Ω, (cid:17) . Note:
All degree theory concepts and results are also valid over any finitedimensional real Hilbert space (such as V or V × V ) instead of R n .2.2 A normalization argumentTo show that the zero set of a map or a system of equations is bounded,we frequently employ the so-called normalization argument. Here, a certainsequence of vectors (with their norms going to infinity) is normalized to yielda unit vector that violates a given criteria. We illustrate this in the followingresult, which will be used later.
Proposition 4
Let A and B two linear transformations on V and p ∈ V .Suppose that h x ⊓ y = 0 , Ax + By = 0 i ⇒ ( x, y ) = (0 , . Xiaoni Chi et al.
Then, the set n ( x, y ) : x ⊓ y = 0 , Ax + By − tp = 0 for some t ∈ [0 , o is bounded. Proof.
Suppose the above set is unbounded. Let z k := ( x k , y k ), t k ∈ [0 ,
1] with || z k || → ∞ , and x k ⊓ y k = 0 , and Ax k + By k − t k p = 0 for all k = 1 , , . . . . Wedivide each of the above equations by || z k || (so as to create normalized vectors z k || z k || ). We let k → ∞ and suppose without loss of generality, x := lim x k || z k || and y := lim y k || z k || . Then, x ⊓ y = 0 , and Ax + By = 0 . However, from || z k || = || x k || + || y k || , we get || x || + || y || = 1, which contradicts ourassumption. The stated conclusion follows. ⊓⊔ -pair In the setting of V = R n , the concepts of LCP-degree of a matrix and HLCP-degree of a pair of matrices are useful in describing the existence and stabilityof solutions, see [6] and [15]. In what follows, we extend these to EuclideanJordan algebras. Consider linear transformations M , A , and B on (a generalalgebra) V and recall thatHLCP( A, B, q ) := wHLCP(
A, B, , q ) and LCP( M, q ) := HLCP( I, − M, q ) . In view of Item ( iii ) in Proposition 1, HLCP(
A, B, q ) is equivalent to finding( x, y ) ∈ V × V such that x ⊓ y = 0 ,Ax + By = q, (6)and LCP( M, q ) is equivalent to finding an x ∈ V such that x ⊓ ( M x + q ) = 0 . We say that M has the R property on V if zero is the only solution ofLCP( M, x ⊓ M x = 0 ⇔ x = 0 . When this condition holds, for any bounded open set Ω in V that containszero, deg( θ, Ω,
0) is well defined, where θ ( x ) := x ⊓ M x.
This common value – which is ind( θ, degree of M , denotedby deg( M ). We now extend this concept to a pair of transformations. eighted complementarity problem 9 Definition 1
The pair { A, B } is said to be an R -pair if zero is the onlysolution of HLCP( A, B, Θ ( z ) = 0 ⇔ z = 0 , where Θ ( z ) := (cid:20) x ⊓ yAx + By (cid:21) with z = ( x, y ). When this condition holds, ind( Θ,
0) is well defined. (Note thatthis equals deg(
Θ, Ω,
0) for any bounded open set Ω in V × V that containszero.) We define the HLCP-degree of the pair { A, B } bydeg( A, B ) := ind( Θ, . Our first result extends Corollary 5.2.6 in [15] from R n to a general Eu-clidean Jordan algebra. Proposition 5
Let M be a linear transformation on V with the R property.Then { I, − M } is an R -pair and deg( I, − M ) = deg( M ) . Proof.
It is easy to see that { I, − M } is an R -pair. Now define the map Θ ( z, t ) := (cid:20) y ⊓ [ tx + (1 − t ) M y ] x − tM y (cid:21) , where z = ( x, y ) and t ∈ [0 , Θ ( z,
0) = (cid:20) θ ( y ) x (cid:21) and Θ ( z,
1) = Θ ( z ) , where θ ( y ) = y ⊓ M y.
Moreover, since M has the R property (so that y ⊓ M y = 0 ⇒ y = 0), it is easy to verify that Θ ( z, t ) = (0 , ∈ V × V if andonly if z = 0. This means that the zero set of Θ ( z, t ) (as t varies over [0 , { (0 , } . Thus, for arbitrary bounded open sets Ω and Ω both containingzero in V , letting Ω = Ω × Ω , we have, by the homotopy invariance of degreeand the Cartesian product formula (see [4], Proposition 2.1.3(h)),deg( I, − M ) = deg (cid:16) Θ ( · , , Ω, (cid:17) = deg (cid:16) Θ ( · , , Ω, (cid:17) = deg (cid:16) θ, Ω , (cid:17) deg (cid:16) I, Ω , (cid:17) = ind( θ,
0) = deg( M ) , where I denotes the identity transformation. ⊓⊔ In the standard LCP theory, R -matrices [2] form an important subclass ofmatrices for which LCP-degree is nonzero. (We note that there are other ma-trices, such as N -matrices of first category satisfying this property [6].) Recallthat M is an R -matrix (in the standard LCP setting) if there is some d > R n such that zero is the only vector that solves the problems LCP( M,
0) andLCP(
M, d ). We now consider a generalization.
Definition 2
Let A and B be two linear transformations on V . We say that { A, B } is an R -pair if it is an R -pair and there exists p ∈ V such that( a ) HLCP( A, B, p ) has a unique solution, say, ( x, y ),( b ) x + y >
0, and( c ) The derivative of G at ( x, y ) is nonsingular, where G ( z ) := (cid:20) x ⊓ yAx + By − p (cid:21) with z = ( x, y ).Note that condition ( c ) above is equivalent to: The derivative of Θ (asgiven in the Definition of R -pair) at ( x, y ) is nonsingular.We elaborate on the R -pair property and give some examples. Suppose { A, B } is an R -pair. As x ⊓ y = 0 and x + y > x and y operator commute(see Proposition 1) and so, with respect to some Jordan frame { e , e , . . . , e n } ,we can write x = k X x i e i and y = n X k +1 y j e j , where 1 ≤ k ≤ n and x i , y j > i and j . Let z := ( x, y ), α := { , , . . . , k } and β = { k + 1 , . . . , n } . (Note that one of these sets may be empty.) Thus, the element x − y = k X x i e i − n X k +1 y i e i is invertible (which means that all the eigenvalues are nonzero). In view ofLemma 19 in [7], the map G defined in condition ( c ) is Fr´echet differentiable.Let Φ ( x, y ) := x ⊓ y = x − Π V + ( x − y ) , where Π V + denotes the projection operator onto V + . Then, the partial deriva-tive of Φ with respect to x at z = ( x, y ) is given by Φ ′ x ( z ) = I x − Π ′ V + ( x − y ) ◦ I x , where I x denotes the identity transformation and ‘ ◦ ′ denotes the composition.Now, we use the formula for the derivative of Π V + given in Lemma 19 of [7].(Although this formula is stated in the setting of a simple Euclidean Jordanalgebra, by writing a general Euclidean Jordan algebra as a product of simpleones, we can show that that the formula is valid in any Euclidean Jordan eighted complementarity problem 11 algebra.) Then, for any h ∈ V with Peirce decomposition h = P n h i e i + P i Let V = R n . Then the standard coordinate vectors form the onlyJordan frame and for any element h ∈ R n , h ij = 0 for all i < j . Thus, in orderto verify the implication (8), we let u = ( u , u , . . . , u k , , , . . . , T and v = (0 , , . . . , , v k +1 , . . . , v n ) T and suppose that ( u, v ) = (0 , x = ( x , x , . . . , x k , , , . . . , T and y = (0 , , . . . , , y k +1 , . . . y n ) T , where x i and y j are positive, we see that forall small ε > 0, the pair ( x + ε u, y + ε v ) is a solution of HLCP( A, B, p ). Thiscontradicts the uniqueness assumption ( a ). Hence condition ( c ) is superfluousin this setting. Example 2 Let M : V → V be a linear transformation that has the R property with respect to V + . This means that for some d > V , zerois the only solution of the linear complementarity problems LCP( M, 0) andLCP( M, d ). We claim that { I, − M } is an R -pair. It is easy to see that the problem HLCP( I, − M, 0) has (0 , 0) as the only solution which means that { I, − M } is an R -pair. Also, HLCP( I, − M, d ) has ( d, 0) as the only solution.This means that with ( x, y ) = ( d, a ) and ( b ) in the abovedefinition are satisfied. We show that condition ( c ) holds. If z = ( x, y ) is closeto ( d, x − y is close to d − 0; hence for all such ( x, y ), Π V + ( x − y ) = x − y and x ⊓ y = x − ( x − y ) = y . Thus, when z is close to ( d, G ( z ) := (cid:20) yIx − M y − p (cid:21) and G ′ ( z ) := (cid:20) II − M (cid:21) . In view of Proposition 3, G ′ ( z ) is nonsingular. Thus, we have verified condition( c ). Hence { I, − M } is an R -pair. Proposition 6 Suppose { A, B } is an R -pair. Then, deg( A, B ) is nonzero. Proof. The pair { A, B } satisfies conditions in Definition 2. Let G ( z, t ) := (cid:20) x ⊓ yAx + By − tp (cid:21) , where z = ( x, y ) and t ∈ [0 , G ( z, 1) = (cid:20) x ⊓ yAx + By − p (cid:21) and G ( z, 0) = (cid:20) x ⊓ yAx + By (cid:21) . As { A, B } is an R -pair, by a normalization argument (see Proposition 4), wesee that the zero sets of G ( z, t ) as t varies are uniformly bounded. Suppose Ω is a bounded open set in V × V that contains all these zero sets. Notethat G ( z, 1) vanishes only at ( x, y ) ∈ Ω and its derivative at this point isnonsingular. Thus,deg( A, B ) = deg (cid:16) G ( · , , Ω, (cid:17) = deg (cid:16) G ( · , , Ω, (cid:17) = sgn det G ′ ( z, = 0 . This proves that deg( A, B ) is nonzero. ⊓⊔ We now discuss the solvability of wHLCP( A, B, w, q ). We recall that ( x, y ) isa solution of wHLCP( A, B, w, q ) if and only if it is a solution of the system x + y − p x + y + 2 w = 0 ,Ax + By − q = 0 . We show that this system has a solution under a nonzero degree condition. Theorem 1 Let { A, B } be an R -pair with deg( A, B ) nonzero. Then for any ( w, q ) ∈ V + × V , wHLCP( A, B, , w, q ) has a nonempty compact solution set. eighted complementarity problem 13 Proof We fix ( w, q ) ∈ V + × V . With z = ( x, y ) ∈ V × V and t ∈ [0 , F ( z, t ) := (cid:20) x + y − p x + y + 2 twAx + By − tq (cid:21) ,H ( z, t ) := " t h x + y − p x + y i + (1 − t ) x ⊓ yAx + By . We show below that there is some bounded open set Ω in V × V which containsall the zeros (in z ) of F and H (as t varies over [0 , Ω , F is ahomotopy connecting F ( z, 1) and F ( z, H is a homotopy connecting H ( z, (cid:16) = F ( z, (cid:17) and H ( z, (cid:16) F ( · , , Ω, (cid:17) = deg (cid:16) H ( · , , Ω, (cid:17) = 0 . This shows that the equation F ( z, 1) = 0 has a nonempty bounded solutionset.To justify these, we proceed as follows.Let Z := { z : F ( z, t ) = 0 for some t ∈ [0 , } . We show by a normalization argument that Z is bounded. Suppose, if possible, Z is unbounded. Let z k := ( x k , y k ), t k ∈ [0 , 1] with || z k || → ∞ , and F ( z k , t k ) =0 for all k = 1 , , . . . . Let k → ∞ and without loss of generality, x := lim x k || z k || and y := lim y k || z k || . We note that || x || + || y || = 1 . Dividing each componentof F ( z k , t k ) by || z k || and letting k → ∞ , we get x + y − p x + y = 0 ,Ax + By = 0 . By Proposition 1, ( x , y ) becomes a nonzero solution of HLCP( A, B, 0) con-tradicting the R property of { A, B } . Hence, Z is bounded.Next, in view of Proposition 2 and the R property of { A, B } , { z : H ( z, t ) = 0 for some t ∈ [0 , } = { (0 , } . Let Ω be a bounded open set in V × V that contains the zero sets of F and H . Then, by the homotopy invariance property of the degree,deg (cid:16) F ( · , , Ω, (cid:17) = deg (cid:16) F ( · , , Ω, (cid:17) = deg (cid:16) H ( · , , Ω, (cid:17) = deg (cid:16) H ( · , , Ω, (cid:17) and deg (cid:16) H ( · , , Ω, (cid:17) = deg (cid:16) Θ, Ω, (cid:17) = ind( Θ, 0) = deg( A, B ) . As the last quantity, by assumption, is nonzero, we conclude thatdeg (cid:16) F ( · , , Ω, (cid:17) = 0 . This means that the equation F ( z, 1) = 0 has a zero in Ω proving the ex-istence of a solution of wHLCP( A, B, w, q ). As all zeros of F ( · , 1) are in thebounded set Ω and the solution set of wHLCP( A, B, w, q ) is clearly closed,we see nonemptyness and compactness of this solution set. This completes theproof. ⊓⊔ Motivated by interior point methods, we consider the case w > 0. First,we make a simple observation: h x ≥ , y ≥ , and x ◦ y > i = ⇒ x > y > . This follows from Item ( iv ), Lemma 2.6 in [18]. Here is a short/different proof.Let x ≥ y ≥ x ◦ y = w > 0. Suppose x > x . This means that there is a primitive idempotent e (whichbelongs to the Jordan frame that appears in the spectral decomposition of x )such that x ◦ e = 0. But then, 0 < h w, e i = h x ◦ y, e i = h y, x ◦ e i = 0, is acontradiction. Hence, x > y > Corollary 1 Let { A, B } be an R -pair with deg( A, B ) nonzero. Suppose w > . Then for any q ∈ V , the following ‘interior point system’ has a nonemptycompact solution set: x > , y > , x ◦ y = w, and Ax + By = q. We now specialize the above two results for a single linear transformation. Corollary 2 Let M be a linear transformation on V . Suppose that M hasthe R property and deg( M ) is nonzero. Then, for all ( w, q ) ∈ V + × V , theweighted linear complementarity problem x ≥ , y ≥ , x ◦ y = w, and y = M x + q has a nonempty compact solution set. In particular, when w > , for any q ∈ V , the ‘interior point system’ x > , y > , x ◦ y = w, and y = M x + q has a nonempty compact solution set. Remarks. In the standard LCP literature, the solvability and uniquenessissues of interior point systems are usually addressed for special types of P -matrices (e.g., P ∗ -matrices), see [9], Lemma 4.3 and Theorem 4.4. In thisregard, the above result appears to be new even in the case of V = R n , asit holds for numerous types of non P -matrices such as strictly copositivematrices, R -matrices, and N -matrices of first category [2]. eighted complementarity problem 15 Fixing the pair { A, B } , we let SOL( w, q ) denote the solution set ofwHLCP( A, B, w, q ). The following result describes the behavior of the map( w, q ) SOL( w, q ). Theorem 2 Suppose { A, B } is an R -pair with deg( A, B ) nonzero. Then, thefollowing statements hold: ( a ) The solution map ( w, q ) SOL( w, q ) from V + × V to V + × V + is uppersemicontinuous. ( b ) Let w k ≥ for all k = 1 , , . . . , and w k → w . Suppose ( x k , y k ) ∈ SOL( w k , q ) for all k . Then, the sequence { ( x k , y k ) } is bounded and any accumulationpoint of this sequence solves wHLCP( A, B, w, q ) . ( c ) Let w > , t k ↓ , and ( x k , y k ) ∈ SOL( t k w, q ) for all k . Then, x k > and y k > for all k , the sequence { ( x k , y k ) } is bounded, and any accumulationpoint of this sequence solves HLCP( A, B, q ) . Proof. ( a ) We fix ( w ∗ , q ∗ ) ∈ V + × V and let Ω be any open set in V × V con-taining SOL( w ∗ , q ∗ ). We show that for all ( w, q ) near ( w ∗ , q ∗ ), SOL( w, q ) iscontained in Ω . Assuming the contrary, suppose there is a sequence { ( w k , q k ) } converging to ( w ∗ , q ∗ ) such that some solution ( x k , y k ) in SOL( w k , q k ) belongsto Ω c (the complement of Ω ). The sequence { ( x k , y k ) } has to be bounded;else, a normalization argument (such as the one used in the previous theorem)produces a nonzero solution of HLCP( A, B, 0) contradicting the R propertyof the pair { A, B } . Now, a subsequential limit of the sequence belongs toSOL( w ∗ , q ∗ ) and at the same time is in Ω c (as this set is closed). This contra-diction proves the upper semicontinuity property of the solution set.( b ) Under the stated assumptions, ( w k , q ) → ( w, q ). A normalization argumentshows that the sequence { ( x k , y k ) } is bounded. Any subsequential limit of thissequence, clearly, belongs to SOL( w, q ), that is, solves wHLCP( A, B, w, q ).( c ) That x k > y k > k follows from Corollary 1. For the remain-ing statements, we specialize ( b ) with w k := t k w . ⊓⊔ V A linear transformation M on V is said to be a P -transformation [7] if x and M x operator commute x ◦ M x ≤ (cid:27) ⇒ x = 0 . P -transformations are generalizations of P -matrices. An important example of P -transformations appears in dynamical systems: the Lyapunov transforma-tion X AX + XA T on the Euclidean Jordan algebra of n × n real symmetricmatrices is a P -transformation if and only if the (real square) A is positivestable. See [7] for properties of P -transformations and further examples. Wenow extend this notion to a pair of transformations. Definition 3 A pair of linear transformations { A, B } is said to be a P -pairover V if x and y operator commute x ◦ y ≤ Ax + By = 0 ⇒ ( x, y ) = (0 , . Below, we collect some properties of such pairs. Proposition 7 Suppose { A, B } is a P -pair. Then, the following statementshold. ( a ) A and B are invertible. ( b ) − B − A and − A − B are P -transformations. ( c ) { A, B } is an R -pair. ( d ) For all ( w, q ) ∈ V + × V , wHLCP( A, B, w, q ) has a nonempty compact so-lution set. Proof. ( a ) If Ax = 0 for some x , then h x and 0 operator commute , x ◦ , Ax + B i ⇒ ( x, 0) = (0 , . This shows that A is invertible. Similarly B is invertible.( b ) Let M := − B − A . If x and M x operator commute and x ◦ M x ≤ 0, then,with y := M x = − B − Ax , we see that: x and y operator commute, x ◦ y ≤ Ax + By = 0. Hence ( x, y ) = (0 , 0) and so x = 0 . Thus, − B − A is a P -transformation. Similarly, − A − B is also a P -transformation.( c ) We now show that { A, B } is an R -pair. First, { A, B } is an R -pair: When x ⊓ y = 0 and Ax + By = 0, by Proposition 2, x and y operator commute ,x ◦ y = 0, and Ax + By = 0; so, ( x, y ) = (0 , 0) by the definition of a P -pair.Now let p := Be , where e denotes the unit element in V . Clearly, (0 , e )is a solution of HLCP( A, B, p ). We show that this is the only solution. Let( x, y ) be any solution of HLCP( A, B, p ) so that x and y operator commute, x, y ≥ 0, and x ◦ y = 0. This implies that x and y − e operator commuteand x ◦ ( y − e ) = − x ≤ 0. Since we also have Ax + B ( y − e ) = 0, by thedefinition of P -pair, ( x, y − e ) = (0 , x, y ) = (0 , e ). Since0 + e = e > 0, we see that conditions ( a ) and ( b ) in Definition 2 hold. We nowverify condition ( c ) in that definition. If ( x, y ) is close to (0 , e ), then x − y isclose to − e and so Π V + ( x − y ) = 0 . In this case, x ⊓ y = x − Π V + ( x − y ) = x .Hence, for all z = ( x, y ) near z := (0 , e ), G ( z ) := (cid:20) xAx + By − Be (cid:21) and G ′ ( z ) := (cid:20) I A B (cid:21) . In view of Proposition 3 and the invertibility of B , G ′ ( z ) is nonsingular.( d ) This follows from Proposition 6 and Theorem 1. ⊓⊔ Example 3 Let V = S n , the Euclidean Jordan algebra of all n × n real sym-metric matrices with h X, Y i := trace ( XY ) and X ◦ Y := XY + Y X . Here, eighted complementarity problem 17 S n + is the ‘semidefinite cone’ consisting of positive semidefinite matrices in S n .We write X (cid:23) X ≻ 0, respectively, to denote elements of V + and itsinterior. Let A be an n × n real matrix which is positive stable (which meansthat every eigenvalue of A has positive real part). Then, the Lyapunov trans-formation X AX + XA T is a P -transformation on S n [7]. Consequently,for any W ≻ 0, the following system has a solution: X ≻ , Y ≻ , X ◦ Y = W, and Y = AX + XA T . While the existence of an X ≻ AX + XA T ≻ X can be found satisfying an additional condition X ◦ ( AX + XA T ) = W where W ≻ X X − BXB T , where B is an n × n real Schurstable matrix (which means that every eigenvalue of B has absolute value lessthan one) [7]. R n We now consider V = R n and prove a uniqueness result. Theorem 3 Let V = R n . Then, the following statements are equivalent: ( a ) { A, B } is a P -pair. ( b ) wHLCP( A, B, w, q ) has a unique solution for every ( w, q ) ∈ R n + × R n .(c) HLCP( A, B, q ) has a unique solution for every q ∈ R n . Proof ( a ) ⇒ ( b ): The solvability of wHLCP( A, B, w, q ) has been addressedin the previous result. We now prove uniqueness. Suppose that ( x , y ) and( x , y ) are any two solutions of wHLCP( A, B, w, q ), i.e., x ≥ , y ≥ x ∗ y = wAx + By = q and x ≥ , y ≥ x ∗ y = wAx + By = q. As w ≥ 0, let α := { i : w i > } and β := { i : w i = 0 } . Then, ( x ∗ y ) i = ( x ∗ y ) i = w i > , ∀ i ∈ α, ( x ∗ y ) i = ( x ∗ y ) i = w i = 0 , ∀ i ∈ β. Now considering the componentwise product (only) over the α indices, we have( x − x ) ∗ ( y − y )= x ∗ y + x ∗ y − x ∗ y − x ∗ y = x ∗ y + x ∗ y − x ∗ (cid:18) x ∗ y x (cid:19) − x ∗ y = ( x − x ) ∗ y + (cid:18) x ∗ y x (cid:19) ∗ ( x − x )= ( x − x ) ∗ (cid:18) x ∗ y x − y (cid:19) = − y x ( x − x ) ≤ . And over the β indices,( x − x ) ∗ ( y − y )= x ∗ y + x ∗ y − x ∗ y − x ∗ y = 0 + 0 − x ∗ y − x ∗ y ≤ . Therefore, we have ( x − x ) ∗ ( y − y ) ≤ ,A ( x − x ) + B ( y − y ) = 0 . As { A, B } is a P -pair and vectors in R n always operator commute, we seethat x = x and y = y . Thus we have uniqueness of solution in anywHLCP( A, B, w, q ).( b ) ⇒ ( c ) : This is obvious by taking w = 0.( c ) ⇒ ( a ) : Suppose ( x, y ) = (0 , 0) with x ◦ y ≤ Ax + By = 0. For q := Ax + + By + = Ax − + By − , we see that ( x + , y + ) and ( x − , y − ) are twodistinct solutions of HLCP( A, B, q ). This contradicts condition ( c ). ⊓⊔ We remark that such a uniqueness result may not prevail over general Eu-clidean Jordan algebras even for P -transformations, see the remarks followingTheorem 14 in [7].The above result, especially Item ( c ), allows us to connect P -pairs to the so-called W property for a pair of matrices, see [16]. Concluding Remarks. In this paper, we have presented some existence anduniqueness results for weighted horizontal linear complementarity problemsover Euclidean Jordan algebras. These are established for R -pairs of lineartransformations satisfying a (nonzero) degree condition. The novelty here isthe use of ‘weighted’ Fischer-Burmeister map and degree theory techniques.We hope to consider applications, algorithms, and non R -pairs in a futurestudy. eighted complementarity problem 19 Acknowledgements The work of the first author is supported by the National NaturalScience Foundation of China (No. 11401126) and Guangxi Natural Science Foundation(Nos. 2016GXNSFBA380102, 2014GXNSFFA118001), China. The third author was sup-ported by Loyola Summer Research Grant 2017. References 1. Anstreicher, K.M.: Interior-point algorithms for a generalization of linear programmingand weighted centering. Optim. Methods Softw. 27(4-5), 605-612 (2012)2. Cottle, R.W., Pang, J.-S., Stone, R.: The Linear Complementarity Problem. AcademicPress, Boston (1992)3. Eisenberg, E., Gale, D.: Consensus of subjective probabilities: the pari-mutuel method.Ann. Math. Statist. 30, 165-168 (1959)4. Facchinei, F., Pang, J.S.: Finite–dimensional variational inequalities and complementar-ity problems. Vol. I, Springer, New York (2003)5. Faraut, J., Koranyi, A.: Analysis on Symmetric Cones. Oxford University Press, NewYork (1994)6. Gowda, M.S.: Applications of degree theory to linear complementarity problems. Math.Oper. Res. 18, 868-879 (1993)7. Gowda, M.S., Sznajder, R., Tao, J.: Some P-properties for linear transformations onEuclidean Jordan algebras. Special issue on Positivity, Linear Algebra Appl. 393, 203-232(2004)8. Kojima, M., Mizuno, S., Noma, T.: Limiting behavior of trajectories generated by acontinuation method for monotone complementarity problems. Math. Oper. Res. 15(4),662-675 (1990)9. Kojima, M., Megiddo., Noma, T., Yoshise, A.: A Unified Approach to Interior PointAlgorithms for Linear Complementarity Problems. Lecture Notes in Computer Science538, Springer-Verlag, Berlin (1991)10. Lloyd, N.G.: Degree Theory. Cambridge University Press, London (1978)11. Ortega, J.M., Rheinboldt, W.C.: Iterative Solutions of Nonlinear Equations in SeveralVariables, Academic Press, New York (1970)12. Ouellette, D.V.: Schur complements and statistics. Linear Algebra Appl. 36, 187-295(1981)13. Potra, F.A.: Weighted complementarity problems – a new paradigm for computingequilibria. SIAM J. Optim. 22(4), 1634-1654 (2012)14. Potra, F.A.: Sufficient weighted complementarity problems. Comput. Optim. Appl. 64:467-488 (2016)15. Sznajder, R.: Degree-theoretic analysis of the vertical and horizontal linear complemen-tarity problems, PhD Thesis, University of Maryland Baltimore County (1994)16. Sznajder, R., Gowda, M.S.: Generalizations of P0