ON STEIN TRANSFORMATION IN SEMIDEFINITE LINEAR COMPLEMENTARITY PROBLEMS

Abstract

In the setting of semidenite linear complementarity problems on $S^n$, we focus on the Stein Transformation $S_A(X)\;:=X-AXA^T$, and show that $S_A$ is (strictly) monotone if and only if ${\nu}_r(UAU^T{\circ}\;UAU^T)$(<)${\leq}1$, for all orthogonal matrices U where ${\circ}$ is the Hadamard product and ${\nu}_r$ is the real numerical radius. In particular, we show that if ${\rho}(A)$ < 1 and ${\nu}_r(UAU^T{\circ}\;UAU^T){\leq}1$, then SDLCP($S_A$, Q) has a unique solution for all $Q{\in}S^n$. In an attempt to characterize the GUS-property of a nonmonotone $S_A$, we give an instance of a nonnormal $2{\times}2$ matrix A such that SDLCP($S_A$, Q) has a unique solution for Q either a diagonal or a symmetric positive or negative semidenite matrix. We show that this particular $S_A$ has the $P^{\prime}_2$-property.

1. Introduction

Given a continuous function f from a real Hilbert space H into itself and a closed convex set K in H, the variational inequality problem VI(f,K) is to find a vector x∗ in H such that

This problem has been extensively studied in the literature. In the infinite dimensional setting, it appears in the study of partial differential equations, mechanics, etc. [17]. In the finite dimensional setting, it appears in optimization, economics, traffic equilibrium problems etc. [13].

Now suppose K is a cone, i.e., tK ⊆ K for all t ≥ 0. Then by putting x = 0 and x = 2x∗, the condition

leads to

If we define the dual cone K∗ of K by

then, (2) and (3) together imply that f(x∗) ∈ K∗.

So when K is a closed convex cone, (1) becomes the problem of finding an x∗ ∈ H such that

This is a cone complementarity problem. This problem and its special cases often arise in optimization (Karush-Kuhn-Tucker conditions), game theory (bimatrix games), mechanics (contact problem, structural engineering), economics (equilibrium in a competitive economy) etc. For a detailed description of these applications, we refer to [16], [6], [5], [20], [21]. In this paper, we focus on the so-called semidefinite linear complementarity problem (SDLCP) introduced by Gowda and Song [8]: Let Sn denote the space of all real symmetric n×n matrices, and be the set of symmetric positive semidefinite matrices in Sn. With the inner product defined by ⟨Z,W⟩ := tr(Z W), ∀Z,W ∈ Sn, the space Sn becomes a Hilbert space. Clearly, is a closed convex cone in Sn. Given a linear transformation L : Sn → Sn and a matrix Q ∈ Sn, the semidefinite linear complementarity problem, denoted by SDLCP(L,Q), is the problem of finding a matrix X ∈ Sn such that

Examples of the semidefinite linear complementarity problem are: the standard linear complementarity problem [4], the block SDLCP [27], and the geometric SDLCP of Kojima, Shindoh, and Hara [18]. For detalils on how to reformulate these as the SDLCP of Gowda and Song (6), we refer to the Ph.D. Thesis of Song (Section 1.3 [22]). We give the description of the standard linear complementarity problem here that is needed in the paper. Consider the Euclidean space Rn with the cone of nonnegative vectors Given a matrix M ∈ Rn×n and a vector q ∈ Rn, the linear complementarity problem LCP(M, q) is to find a vector in Rn such that

where ⟨x, y⟩ is the usual inner product in Rn. This problem has been studied in great detail, see [4], [6], [5], [20]. In this setting, we have the following result:

where yi denotes the ith element of the vector y.

As mentioned above, the standard linear complementarity problem is a special case of the semidefinite linear complementarity problem in the following way [22]:

Let a matrix M ∈ Rn×n and a vector q ∈ Rn be given. Define a linear transformation M : Sn → Sn by M(X) := Diag(Mdiag (X)), where diag(X) is a vector whose entries are the diagonal entries of the matrix X, and Diag(u) is a diagonal matrix whose diagonal is the vector u. Corresponding to LCP(M, q) in (7), one can consider SDLCP(M,Diag(q)), which is to find X ∈ Sn such that If X is a solution of SDLCP(M,Diag(q)), then diag(X) solves LCP(M, q). Conversely, if x solves LCP(M, q), then Diag(x) solves SDLCP(M,Diag (q)). In this sense, these two complementarity problems are equivalent.

We want to note that the cone of nonnegative vectors in Rn is polyhedral. That is, is the intersection of a finite number of sets defined by linear inequalities. However, the cone of symmetric positive semidefinite matrices in Sn is not polyhedral. We want to note that because of the nonpolyhedrality of the cone and the noncommutativity of the matrix product, the results available for the standard linear complementarity problem do not simply carry over to the SDLCPs. To ellaborate on this, for example, consider the so-called the P and GUS-properties introduced by Gowda and Song [8]: A linear transformation L : Sn → Sn has the

Note that (c) is an analogus definition to (b) and in the LCP case, P ⇔ GUS (see (a)). Does the equivalence still hold for SDLCPs? Authors [7], [1], [12] specialized these properties to the Lyapunov transformation LA(X) := AX + XAT, the Stein transformation SA(X) := X − AXAT, the two-sided multiplication transformation MA(X) := AXAT, and studied interconnections with the (global) asymptotic stability of the continuous and discrete linear dynamical systems (see Introduction of [12])

Gowda and Song showed that (Thm 5 [8]) the Lyapunov transformation LA(X) := AX + XAT has the P-property if and only if A is positive stable (i.e., every eigenvalue of A has a positive real part) and hence the continuous system in (8) is globally asymptotically stable [19]; and has (Thm 9 [8]) the GUS-property if and only if A is positive stable and positive semidefinite. Therefore, the P and GUS properties are not equivalent in SDLCP setting.

Bhimasankaram et al. [1] showed that for the two-sided multiplication transformation MA(X) := AXAT, GUS and P are both equivalent to A being positive definite or negative definite. Zhang [28], in particular, looked at a problem of solving the matrix equation AXAT + BY BT = C seeking a solution X ⪰ 0 for given matrices A,B and C and provided necessary and sufficient conditions.

Gowda and Parthasarathy (Theorem 11, Remark 4 [7]) showed that the Stein transformation SA(X) := X−AXAT has the P-property if and only if ρ(A) < 1 and hence the discrete system in (8) is globally asymptotically stable [23]. However, the characterization of the GUS-property of the Stein transformation is still open. The known results so far are in 2003, Gowda, Song, and Ravindran (Thm 3 [11]) showed that if SA is strictly monotone (that is, ⟨X; SA(X)⟩ > 0 for all 0 ≠ X ∈ Sn), then SA has the GUS-property; If A is normal, the converse also holds. Moreover, in 2013, J. Tao [26] examined conditions which gives the so-called Cone-GUS-property (that is, SDLCP(SA,Q) has a unique solution for all .

Therefore, the aims of this paper are to give a characterization of SA being strictly monotone in terms of the real numerical radius of A and hence providing a sufficient condition for the GUS-property of SA; and to examine a particular instance of A so that SDLCP(SA,Q) has a unique solution for all Q diagonal or symmetric negative semidefinite. The main results of the paper are as follows: In section 2, we show that the Stein Transformation SA is (strictly) monotone if and only if vr(UAUT ◦ UAUT) (<) ≤ 1 for all U orthogonal, where ◦ denotes the Hadamard product and vr denotes the real numerical radius defined in this paper (see Section 2). In particular, if ρ(A) < 1 and vr(UAUT ◦ UAUT) ≤ 1 then SDLCP (SA,Q) has a unique solution for all Q ∈ Sn. As a by-product, we get a result that if tr(AAT) > n, then SA is not monotone. In section 3, we look at a particular case of the 2 × 2 matrix A (motivated by [10]):

and show that SDLCP(SA,Q) has a unique solution if Q is either a diagonal or a symmetric negative semidefinite matrix. Moreover, we show that not only SA but also every principal subtransformations of defined by invertible (X ∈ Sn) has the Cone-Gus-properties.

We list below some necessary definitions:

(d) The Hadamard product (or Schur product) of two matrices A and B is the entrywise product of A and B.

 

2. Characterization of the Monotonicity of the Stein Transformation

Recall [14] that the Numerical Radius of an n × n matrix A is defined as v(A) = max{|xTAx| : ∥x∥ = 1, x ∈ Cn}, where ∥x∥ denotes the Euclidean norm of the vector x. Here we define the real version of the Numerical Radius for a real matrix A ∈ Rn×n as

and relate it with the monotonicity of the Stein Transformation SA.

Example For r > 0, let

Since xTAx = 0 for all x, v(A) = vr(A) = 0.

Note that for a square matrix A, ρ(A) ≤ v(A) since each square matrix A has an eigenvector in Cn, but this inequality is not necessarily true for vr(A). See in the above example that and therefore ρ(A) = r > vr(A), where ρ(A) denotes the spectral radius of A and σ(A) is the spectrum of A (i.e., the set of all eigenvalues of A). We also note that for a square matrix A, because |xTAx| = |xTATx| and is a symmetric matrix.

We characterize the monotonicity of SA in terms of the real numerical radius.

Theorem 2.1. For A ∈ Rn×n, the Stein Transformation SA : Sn → Sn, SA(X) := X − AXAT is (strictly) monotone if and only if for all orthogonal matrices U, vr(UAUT ◦ UAUT) (<) ≤ 1.

Proof. Suppose SA is monotone. Then ⟨SA(D);D⟩ ≥ 0, where D is a diagonal matrix with the diagonal equals a unit vector d. Note that

and hence vr(A◦A) ≤ 1. Since SA is monotone, SUAUT is also monotone for all orthogonal matrices U. Therefore, vr(UAUT ◦ UAUT) ≤ 1. For the converse, let B = UAUT and suppose vr(B ◦ B) ≤ 1 for all orthogonal matrices U. Take X ∈ Sn, then X = UDUT where D is a diagonal matrix with the diagonal d. Upon replacing X by UDUT and carrying out the calculation we get

as tr(BDBTD) = ⟨d, (B ◦ B)d⟩. By our assumption vr(B ◦ B) ≤ 1, therefore ⟨SA(X),X⟩ ≥ 0 for all X ∈ Sn. Hence SA is monotone.

The proof for strict monotonicty is similar

We now state a simple check condition for a nonmonotonicity of SA below.

Corollary 2.2. If tr(AAT) > n for A ∈ Rn×n, then SA is not monotone.

Proof. We shall show that if SA is monotone, then tr(AAT) ≤ n.

First we observe that for a diagonal matrix D,

Therefore, if SA is monotone, (I−A) ◦ (I+A) is positive semidefinite, as well as (I−B) ◦ (I+B) (for SA monotone implies SB monotone) for all B = UAUT, where U is an arbitrary orthogonal matrix.

Suppose SA is monotone. Then (I−A) ◦ (I+A) = I−A◦A is positive semidefinite which means ⟨e, (I−A◦A)e⟩ ≥ 0, where e is the vector of all 1’s. This reduces to ⟨e, e⟩−⟨e, (A◦A)e⟩ ≥ 0. Since ⟨e, e⟩ = n and ⟨e, (A◦A)e⟩ = tr(AAT), we get the desired result.

We state below a sufficient condition for SA to be GUS in terms of the matrix A.

Theorem 2.3. For A ∈ Rn×n, if p(A) < 1, vr(UAUT ◦ UAUT) ≤ 1 for all orthogonal matrices U, then SA has the GUS-property.

Proof. The condition ρ(A) < 1 is equivalent to SA having the P-property which is also equivalent to the existence of the solution to SDLCP(SA,Q) for all Q ∈ Sn. (Thm 11, Remark 4 [7]). By Theorem 2, SA is monotone. Suppose there are two solutions X1 and X2 to SDLCP (SA,Q). Let Yi = SA(Xi) + Q, i = 1, 2. Thus, 0 ≤ ⟨SA (X1−X2), X1−X2⟩ = ⟨Y1−Y2,X1−X2⟩ ≤ 0 since tr(X1Y1) = 0 = tr (X2Y2) and Xi, Yi ⪰ 0 for i = 1, 2. This leads to tr(X2Y1) + tr (X1Y2) = 0. Since the involved matrices are all positive semidefinite, each trace is nonnegative. Hence tr(X2X1) = 0 = tr (X1Y2), resulting X2Y1 = 0 = X1Y2. Hence SA has the Cross Commutative-property. It is known (Thm 7 [8]) that P + CrossCommutativity ⇔ GUS, hence the proof is complete.

Remark 2.2. As it is seen in the above proof, monotonicity implies the Cross Commutative-property. Whether monotonicity is equivalent to the Cross Commutative-property is an open problem. This would complete the characterization of the GUS-property of SA.

 

3. On a special SA : S2 → S2

In what follows, let RC(X) := Diag(Cxd)+X0, where C ∈ Rn×n is a matrix, xd is the vector composed of the main diagonal of the matrix X, and X0 is the matrix obtained after replacing all the diagonal elements of X with zeros. If we let and C = I − A ◦ A, then

Note that this A is not normal. Motivated by the question raised in (p12, Problem 2. [10]), we studied this particular instance of SA. So far, the following is known:

In [10], Gowda and Song raised the question:

We were able to show that this SA has the GUS-property regardless of the value of ϒ if the given Q is either diagonal or symmetric negative semidefinite. We start with a lemma which shows that every positive definite solution of an SDLCP(L,Q) is locally unique if L has the P-property.

Lemma 3.1. Suppose a linear transformation L : Sn → Sn has the P-property. Then SDLCP(L,Q) cannot have two distinct positive definite solutions.

Proof. Suppose X1 and X2 are two distinct positive definite solutions to a given SDLCP(L,Q). Then we get X1(L(X1) + Q) = 0 = X2(L(X2) + Q): Since both X1 and X2 are invertible, this would mean L(X1) = −Q = L(X2). Hence (X1−X2)L (X1−X2) = 0. By the P-property of L, X1 = X2 contradicting our assumption.

Now, let’s consider our special SA given in (9). Since all eigenvalues are zeros for any ϒ, note that our SA has the P-property for all ϒ.

Theorem 3.2. Let

where ϒ a real number. Then

Proof. For part(a), let Q = D, where D is a diagonal matrix. By Proposition 3(ii) of [10], the solution set of SDLCP(SA,D) is equal to the solution set of LCP(C, d), where C = I − A ◦ A, under obvious modifications (here, d denotes the vector diag(D)). Since C is a P-matrix (that is, the matrix with all its principal minors positive) for all ϒ ∈ R, the assertion follows.

For part(b), if given Q is diagonal, then there is a unique solution by part(a). Assume Q is a nondiagonal symmetric negative semidefinite matrix. Since −Q ⪰ 0, we claim that there is a unique X ≻ 0 such that SA(X) = −Q. This is because, for

and is a linear transformation from Sn to Sn that maps a nondiagonal positive semidefinite matrix into a positive definite matrix; and Therefore, Note X solves SDLCP(SA,Q). Now suppose there is X1 ⪰ 0 such that Y1 := SA(X1) + Q ⪰ 0 and X1Y1 = 0. Then SA(X1) = Y1 − Q and

because Y1 ⪰ 0. Then both X1 and X are positive definite solutions and by Lemma 7, X1 = X. This completes the proof.

In addition to this, in what follows, we show that not only this SA, but also its variants P ∈ Rn×n invertible (X ∈ Sn), all have the Cone-Gus-properties when ϒ2 < 4. We achieve this by showing SA has the so-called P2′-property and then applying results of J. Tao [26]. The P2′-property was introduced by Chandrashekaran et. al [2] in 2010, and J. Tao showed that (Thm 3.3 [26] interpreted for V = Sn) L has the P2′-property if and only if and all its principal subtransformations have the Cone-Gus properties where P ∈ Rn×n invertible (X ∈ Sn).

Theorem 3.3. The transformation SA given in (9) has the P2′-property for ϒ2 < 4.

Proof. Let

Suppose XSA(X)X ⪯ 0. If X is invertible (i.e., X ≻ 0), then

since X−1 is symmetric. Let Q := SA(X) ⪯ 0. Then by Theorem 3(b), SDLCP(SA,Q) has a unique solution and the solution is which equals to by linearity of SA. But this equals −X which is a solution to SDLCP(SA,Q), and so 0 ≻ −X ⪰ 0 which is a contradiction. Therefore, X can’t be invertible.

Now suppose det X = 0, i.e., Carrying out the algebra,

Then XSA(X)X ⪯ 0 leads to [XSA(X)X]11 ≤ 0 and [XSA(X)X]22 ≤ 0. Upon putting in [XSA(X)X]22, we get

because X ⪰ 0 and ϒ2 < 4. Hence the last item in the above equations vanishes, therefore, we have cases

(i) x3 = 0, or (ii) (x1−x3)2+(4−ϒ2)x1x3 = 0. For the case (i), if x3 = 0, then x2 = 0. Then [XSA(X)X]11 leads to Hence x1 = 0, and therefore X = 0. In case (ii), we get 0 ≤ (x1−x3)2 = −(4 − ϒ2)x1x3 ≤ 0, which leads to [x1 = x3] and [x1 = 0 or x3 = 0] since ϒ2 ≠ 4. We get X = 0 in each case. Therefore, SA has the P′2-property.

Hence, for any real invertible matrix P, L(X) := PT SA(PXPT)P has only the trivial solution for any given Q ⪰ 0 which would make X ⪰ 0, L(X)+Q ⪰ 0, and tr(X(L(X)+Q)) = 0, when ϒ2 < 4: From the above proof, it is more or less clear that SA is not P′2 if ϒ2 > 4. When ϒ2 = 4, the following example shows that SA is not P′2 as well.

Example Let

Then XSA(X)X = 0 ⪯ 0, yet X≠ 0. Hence SA is not P′2.

 

4. Conclusion and Acknowledgements

In this paper, we took a baby step to characterize the GUS-property of the Stein transformation which hasn’t been succeeded for the last decade. We hope the presentation of this paper would bring interests of many other talented mathematicians to work on this newly generated problem. We are indebted to Professor Muddappa S. Gowda of University of Maryland Baltimore County for the idea of the real numerical radius and helpful suggestions. We are grateful to Professor Jiyuan Tao of Loyola University of Maryland for pointing out right resources for the paper.