Linear complementarity on simplicial cones and the congruence orbit of matrices
aa r X i v : . [ m a t h . O C ] O c t Linear complementarity on simplicial cones and the congruenceorbit of matrices ∗ A. B. N´emethFaculty of Mathematics and Computer ScienceBabe¸s Bolyai University, Str. Kog˘alniceanu nr. 1-3RO-400084 Cluj-Napoca, Romaniaemail: [email protected]. Z. N´emethSchool of Mathematics, University of BirminghamWatson Building, EdgbastonBirmingham B15 2TT, United Kingdomemail: [email protected]
Abstract
The congruence orbit of a matrix has a natural connection with the linear complementarity problemon simplicial cones formulated for the matrix. In terms of the two approaches – the congruence orbitand the family of all simplicial cones – we give equivalent classification of matrices from the point ofview of the complementarity theory.
1. Introduction
We use in this introduction some standard terms and notations which will also be spec-ified in the next section.Let be K ⊂ R m a cone, K ∗ ∈ R m be its dual, M : R m → R m a linear mapping and q ∈ R m . The problemLCP( K, q, M ) : find x ∈ K with M x + q ∈ K ∗ and h x, M x + q i = 0is called the linear complementarity problem on the cone K . In the case K = R m + it isdenoted by LCP( q, M ) and called the classical linear complementarity problem .As one of the most important problems in optimization theory, the classical linearcomplementarity problem has a broad literature (see [1] and the literature therein). ∗ Key words and phrases: linear complementarity, Q -matrix, P -matrix,congruence orbit espite of the important progress last decades in this field, it still in the center of interestnowadays.Besides the classical case, the linear complementarity on Lorentz cone and the cone ofpositive semi-definite matrices emerged as an important topic in the previous decade [2,3].When K ⊂ R m is a simplicial cone, the linear complementarity problem can betransformed by a linear mapping in the classical one. But general simplicial cones candiffer substantially from each other in some aspects, one of them being the projectiononto the cone, the mapping playing an essential role in the solution of optimizationproblems. If the linear mapping M is given, such an approach relates the problem to the congruence orbit of M , i.e. to the set of maps O ( M ) = { L ⊤ M L : L ∈ GL(m , R ) } , (1)where GL(m , R ) denotes the general linear group of R m , i.e., the group of invertible linearmaps of the vector space R m .Among other results, in this note we will show that LCP( K, q, M ) is feasible for anarbitrary simplicial cone K ⊂ R m and an arbitrary q ∈ R m if and only if M is a positivedefinite mapping [1], i.e., if h M x, x i > ∀ x ∈ R m , x = 0, which is equivalent to sayingthat this property holds for each member of O ( M ). It turns out that this property isalso equivalent with the much stronger P and Q -properties of all members of O ( M ) andequivalently, with the corresponding properties of M for each simplicial cone K .It is possible that some of the problems considered in the present note already occuredin a different setting in the huge literature on linear complementarity. Even so, theapproach of considering classical P and Q -properties of a matrix for the whole family ofsimplicial cones and the relation with the congruence orbit of the matrix seems a novelapproach which justifies our following investigation.
2. Terminology and notations
Denote by R m the m -dimensional Euclidean space endowed with the scalar product h· , ·i : R m × R m → R , and the Euclidean norm k · k and topology this scalar productdefines. Denote h x, y i = 0 by x ⊥ y .Throughout this note we shall use some standard terms and results from convexgeometry (see e.g. [4]).Let K be a convex cone in R m , i. e., a nonempty set with (i) K + K ⊂ K and (ii) tK ⊂ K, ∀ t ∈ R + = [0 , + ∞ ). The convex cone K is called pointed , if K ∩ ( − K ) = { } . The cone K is generating if K − K = R m . K is generating if and only if int K = ∅ . A closed, pointed generating convex cone is called proper cone .The set K = cone { x , . . . , x m } := { t x + · · · + t m x m : t i ∈ R + , i = 1 , . . . , m } (2) ith x , . . . , x m linearly independent vectors in R m is called a simplicial cone . A sim-plicial cone is proper.The dual of the convex cone K is the set K ∗ := { y ∈ R m : h x, y i ≥ , ∀ x ∈ K } , with h· , ·i is the standard scalar product in R m .The cone K is called self-dual if K = K ∗ . If K is self-dual, then it is proper.In all that follows we will suppose that R m is endowed with a Cartesian system havingan orthonormal basis e , . . . , e m and the elements x ∈ R m are represented by the columnvectors x = ( x , . . . , x m ) ⊤ , with x i the coordinates of x with respect to this basis. (Thatis, R m will be the vector space of m -dimensional column vectors.)The set R m + = { x = ( x , . . . , x m ) ⊤ ∈ R m : x i ≥ , i = 1 , . . . , m } is called the non-negative orthant of the above introduced Cartesian reference system.In fact R m + = cone { e , ..., e m } . A direct verification shows that R m + is a self-dual cone.If K is the simplicial cone defined by (2) and A is the non-singular matrix transformingthe basis e , ..., e m to the linear independent vectors x , ..., x m , then obviously K = A R m + . (3)For simplicity from now on we will call a convex cone simply cone.
3. Changing the cone linearly
Lemma 1
Let W ⊂ R m be a cone and A ∈ GL(m , R ) . Then K = AW is a cone tooand K ∗ = A − T W ∗ . Proof.
The first assertion is trivial.Take y ∈ K ∗ . This is equivalent to h Aw, y i = h w, A ⊤ y i ≥ , ∀ w ∈ W. Thus, y ∈ K ∗ ⇐⇒ A ⊤ y ∈ W ∗ ⇐⇒ y ∈ A − T W ∗ . ✷ Corollary 1 If W ∗ = W , A ∈ GL(m , R ) then K = AW ⇐⇒ K ∗ = A − T W. If K is the simplicial cone (2), then, because of the representation (3) and the self-dualityof R m + , we have K ∗ = A − T R m + . . Linear transformation of a cone and the complementarityproblem For the mapping F : R m → R m the complementarity problem CP(
F, K ) is to find x ∈ R m such that K ∋ x ⊥ F ( x ) ∈ K ∗ . The solution set of CP(
F, K ) will be denoted by SOL(
F, K ). We have x ⊥ y ⇐⇒ Ax ⊥ A −⊤ y. (4)Hence, by using Lemma 1 and (4), we conclude the following result: Proposition 1 If W is a cone, A ∈ GL(m , R ) and K = AW , then SOL(
F, K ) = A (SOL( A ⊤ F A, W )) . Proof.
Indeed, Ax ∈ A (SOL( A ⊤ F A, W )) ⇐⇒ x ∈ SOL( A ⊤ F A, W ) ⇐⇒ W ∋ x ⊥ A ⊤ F ( Ax ) ∈ W ∗ ⇐⇒ K ∋ Ax ⊥ F ( Ax ) ∈ K ∗ ⇐⇒ Ax ∈ SOL ( F, K ) . ✷
5. The case of linear complementarity
The complementarity problem CP( f, K ) with F ( x ) = M x + q , where M ∈ R m × m and q ∈ R , will be denoted by LCP( K, M, q ) and called linear complementarity problem .Thus, the linear complementarity problem LCP(
K, M, q ) is to find x ∈ R m such that K ∋ x ⊥ M x + q ∈ K ∗ . The solution set of LCP(
K, M, q ) will be denoted by SOL(
K, M, q ). In this caseProposition 1 becomes
Proposition 2 If W is a cone, A ∈ GL(m , R ) and K = AW , then SOL(
K, M, q ) = A (SOL( W, A ⊤ M A, A ⊤ q )) . If K = R m + , then CP( F, K ), SOL(
F, K ), LCP(
K, M, q ) and SOL(
K, M, q ) will simplybe denoted by CP( F ), SOL( F ), LCP( M, q ) and SOL(
M, q ), respectively. . The congruence orbit of a matrix and the complementarityproblem If A and B are in R m × m , then they are congruent and we write A ∼ B , if there exists L ∈ GL(m , R ) such that L ⊤ AL = B, that is B is in the congruence orbit O ( A ) of A defined at (1). Obviously, ∼ is anequivalence relation and in this case O ( B ) = O ( A ) . In the case of simplicial cones Proposition 2 reduces to
Proposition 3 If K = L R m + is a simplicial cone then SOL(
K, A, q ) = A (SOL( R m + , L ⊤ AL, A ⊤ q )) . Hence the linear complementarity problem on a simplicial cone is equivalent to the com-plementarity problem on the non-megative orthant, that is, to the classical linear com-plementarity problem.
Remark 1
Proposition 3 shows that for linear complementarity problems with matrix A on simplicial cones the congruence orbit O ( A ) of A appears in a natural way. Definition 1
Let A be a linear transformation. Then, we say that1. A has the K - Q -property if LCP(
K, A, q ) has a solution for all q .2. A has the K - P -property if LCP(
K, A, q ) has a unique solution for all q .3. The R m + - Q -property ( R m + - Q -property) is called Q -property, ( P -property) and thematrix of the linear transformation A with the Q -property ( P -property) is called Q -matrix ( P -matrix) (for simplicity we denote a linear transformation and its matrixby the same letter).4. A has the general feasibility property with respect to K denoted K - FS -property if ( AK + q ) ∩ K ∗ = ∅ , ∀ q ∈ R m . If K = R m + , then the K - FS -property is called FS -property and it is characterized bythe relation ( A R m + + q ) ∩ R m + = ∅ , ∀ q ∈ R m . The matrix A with the FS -property is called FS -matrix. emark 2 Obviously, the P -property of a matrix A implies its Q -property, and its Q -property implies its FS -property as well. A classical result going back to the paper [5]asserts that A possesses the P -property if and only if all its principal minors are positive.Theorem 3.1.6 in [1] asserts that a positive definite matrix possesses the P -property.The FS -property of a matrix can be considered the weakest one in the context of linearcomplementarity. It is easy to see that the FS -property ( K - FS -property) of the matrix A is equivalent to − A R m + + R m + = R m ( − AK + K ∗ = R m ). With the notations in the above definition we have
Proposition 4 If A ∈ R m × m has the K - Q -property ( K - P -property) then M = L ⊤ AL ∈O ( A ) has the LK - Q -property ( LK - P -property). Example 1
The congruence orbit of the identity
We have O ( I ) = { L ⊤ L : L ∈ GL(m , R ) } .Hence, each member of O ( I ) is a symmetric positive definite matrix . The following lemma is based on Example 1 and shows that if the congruence orbit ofa matrix contains a positive definite matrix, then all of its matrices are positive definite.
Lemma 2 If O ( A ) contains a symmetric positive definite matrix, then O ( A ) = O ( I ) . Proof.
We can suppose that A itself is a symmetric positive definite matrix.If we denote by R the square root of A [6], then we can write O ( A ) = { L ⊤ AL : L ∈ GL(m , R ) } = { ( RL ) ⊤ ( RL ) : L ∈ GL(m , R ) } = { M ⊤ M : M ∈ GL(m , R ) } = O ( I ) . ✷ Obviously, a symmetric positive definite matrix is nothing else but a symmetric P -matrix. Hence, by Lemma 2 we conclude Corollary 2
Each member of the congruence orbit of a symmetric positive definite ma-trix is a P -matrix. How about the congruence orbit of a non-symmetric P -matrix? Can it have a propertysimilar to the one stated in Corollary 2? We will show that this holds if and only if thematrix is positive definite. Lemma 3
If the diagonal of the matrix A ∈ R m × m contains some non-positive element,then O ( A ) contains non- FS -matrices. roof. (a) Let D k be a diagonal matrix with d kk = − d ii = 1 if i = k . If A = ( a ij ) i,j =1 ,...,m , then B = D ⊤ k AD k is a matrix with b ij = a ij if i = k and j = k , b ki = − a ki , i = k , b jk = − a jk , j = k , and b kk = a kk .(b) Without loss of generality we can assume that a ≤
0. Assume that the positiveterms of the first line of A are a j , a k , , ..., a l . Then if D is the diagonal matrix with d ii = − , i ∈ { j, k, ..., l } and d ii = 1 , i / ∈ { j, k, ..., l } , then using the remark at (a) weconclude that C = D ⊤ AD is a matrix whose first line contains only non-positive elements. Hence, for any q =( − , ∗ .... ∗ ) ⊤ we have ( C R m + + q ) ∩ R m + = ∅ . ✷ As we have stated at Remark 2 a positive definite matrix possesses the P -property,but simple examples show, that the converse is not true.We have the obvious assertion: Lemma 4 If A ∈ R m × m is not positive definite, then its symmetrizant S ( A ) = A + A ⊤ is not positive definite neither. Remark 3
Observe that the diagonal of S ( A ) defined as in (5) coincides with the diag-onal of A . Theorem 1
Let A ∈ R m × m . Then the following assertions are equivalent:1. Each member of O ( A ) is a FS -matrix.2. Each member of O ( A ) is a Q -matrix.3. Each member of O ( A ) is a P -matrix.4. A is a positive definite matrix.Hence, if any of the above conditions hold then A possesses the K - P , K - Q , K - FS -properties for any simplicial cone K . Proof.
Suppose that assertion 1 hold.(a) Assume that A is not positive definite, that is, item 4. does not hold. Then,by Lemma 4, the same is true for S ( A ). That is, S ( A ) is a symmetric matrix whichis not positive definite. Then, it has non-positive eigenvalues, that is, in the spectraldecomposition S ( A ) = ODO ⊤ is a diagonal matrix with some non-positive elements. On the other hand we have D = O ⊤ S ( A ) O = O ⊤ AO + O ⊤ A ⊤ O . Now, since O ⊤ A ⊤ O = ( O ⊤ AO ) ⊤ , it follows, according to Remark 3, that the diagonalof D coincides with the diagonal of O ⊤ AO . Since this diagonal contains non-positiveelements, it follows by Lemma 3 that the orbit O ( A ) contains elements which are not FS -matrices. This shows the implication 1 ⇒ L ∈ GL(m , R ) and any x = 0 we have h L ⊤ ALx, x i = h ALx, Lx i > , that is, L ⊤ AL is positive definite and hence, by Theorem 3.1.6 in [1] is a P -matrix, andhence also a Q -matrix and an FS -matrix. Thus, we have the implications 4 ⇒ ⇒ ⇒
1. (c) The last assertion of the theorem follows from Propositions 3 and 4. ✷ Lemma 5
If for A ∈ GL(m , R ) there exists x ∈ R m with h Ax, x i > , then O ( A ) contains a matrix with at least one positive element in its diagonal. Proof.
Let S ( A ) be the symmetrizant of A . Then h S ( A ) x, x i > S ( A ) = ODO ⊤ of S ( A ). Then, h S ( A ) x, x i = h ODO ⊤ x, x i = h DO ⊤ x, O ⊤ x i > D must contain positive elements, since h Dy, y i = P i d ii y i > y = O ⊤ x .Now, D = O − A ( O ⊤ ) − + O − A ⊤ ( O ⊤ ) − O − A ( O ⊤ ) − + ( O − A ( O ⊤ ) − ) ⊤ , hence the diagonal of O − A ( O ⊤ ) − coincides with the diagonal of D and hence it mustcontain positive elements. ✷ The matrix A = ( a ij ) i,j =1 ,...,m ∈ R m × m is called positive , if a ij > ∀ i, j . Lemma 6
For the matrix A ∈ R m × m the following two assertions are equivalent:1. O ( A ) contains a matrix with a positive element on its diagonal, . O ( A ) contains a positive matrix. Proof. (a) We first prove that if the matrix A := ( a ij ) i,j =1 ,....,m ∈ R m × m contains apositive principal submatrix of order n − < m , then it has a conjugate containing apositive principal submatrix of order n .Suppose that A (1 : n, n ) := ( a ij ) i,j =1 ,....,n has the property that a ij > i, j ∈ { , . . . , n } and let A (2 : n, n ) = ( a ij ) i,j =2 ,...,n .Denote by I ∈ R n × n the unit matrix and let E ∈ R n × n be the matrix with 1 in theposition ( i, j ) = (1 ,
2) and 0 elsewhere. Let L t = I + tE with t a real parameter. Put B = L t A (1 : n, n ) L ⊤ t = ( b ij ) i,j =1 ,...,n and B (2 : n, n ) = ( b ij ) i,j =2 ,...,n . Then, we have B (2 : n, n ) = A (2 : n, n ) , (6) b = a + ta + ta + t a , (7) b i = a i + ta i , i ≥ , (8) b i = a i + ta i , i ≥ . (9)From (6), (7), (8) and (9), it follows that for t > b ij > , i, j = 1 , ..., n .(b) Applying the procedure from (a), the element a ii > A can beaugmented to obtain in O ( A ) a matrix with positive principal minor of order 2, then amatrix with positive principal minor of order 3 in O ( A ), and so an, to obtain a positivematrix in O ( A ). ✷ Theorem 2
If for A ∈ GL(m , R ) there exists an x ∈ R m with h Ax, x i > and an y ∈ R m \ { } with h Ay, y i ≤ , then O ( A ) contains non- FS -matrices and Q -matrices aswell. Proof.
By Theorem 1 O ( A ) must contain non- FS -matrices.By Lemma 5, O ( A ) must contain a matrix B with at least one positive element onits diagonal. Then, by Lemma 6, O ( B ) = O ( A ) must contain a positive matrix. ByTheorem 3.8.5 in [1], it follows that a such matrix is a Q -matrix. ✷ Corollary 3
Suppose that A ∈ GL(m , R ) . Then, exactly one of the following alternativeshold:1. h Ax, x i > , ∀ x ∈ R m \ { } ⇐⇒ each member of O ( A ) is a FS -matrix ⇐⇒ eachmember of O ( A ) is a Q -matrix ⇐⇒ each member of O ( A ) is a P -matrix. . If h Ax, x i ≤ , ∀ x ∈ R m , then no matrix in O ( A ) can have the FS -property.3. If for some non-zero elements x and y in R m one has h Ax, x i > and h Ay, y i ≤ ,then O ( A ) contains non- FS -matrices and Q -matrices as well. Proof.
Only item 2 needs proof. Let A ∈ GL(m , R ) a matrix with h Ax, x i ≤ ∀ x ∈ R m .Suppose to the contrary that there is a matrix L ⊤ AL ∈ O ( A ) which has the FS -property.Let q ∈ R m be a vector with all components negative. Then, there exists x ∈ R m + suchthat L ⊤ ALx + q ∈ R m + . (10)Thus, 0 ≤ h x, L ⊤ ALx + q i = h Lx, ALx i + h x, q i . (11)If x = 0, then the right hand side of equation (11) is negative, which is a contradiction.Hence, x = 0. Then, (10) implies q ∈ R m + , which contradicts the choice of q . In conclu-sion, no matrix in O ( A ) can have the FS -property. ✷ The alternatives listed in Corollary 3 can be formulated in complementarity termstoo:
Corollary 4
Suppose that A ∈ GL(m , R ) . Then exactly one of the following alternativeshold :1. A possesses the K - FS - property for any simplicial cone K ⇐⇒ A possesses the K - Q - property for any simplicial cone K ⇐⇒ A possesses the K - P - property forany simplicial cone K .2. There is no simplicial cone K for which A possesses the K - FS -property.3. There exists a simplicial cone K for which A does not have the K - FS -property andthere exists a simpicial cone L for which A possesses the L - Q -property. References [1] R. W. Cottle, J.-S. Pang, and R. E. Stone,
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