Yoon Song

Soongsil University · Department of Mathematics

Given a proper cone K in a finite dimensional real Hilbert space H, we present some results characterizing \(\mathbf{Z}\)-transformations that keep K invariant. We show for example, that when K is irreducible, nonnegative multiples of the identity transformation are the only such transformations. And when K is reducible, they become ‘nonnegative di...

Let V be a Euclidean Jordan algebra with a symmetric cone K. We show that for a Z-transformation L with the additional property L(K) ? K (which we will call ?cone-preserving?), GUS ? strictly copositive on K ? monotone + P. Specializing the result to the Stein transformation SA(X) :

In the setting of semidefinite linear complementarity problems on S n , we focus on the Stein Transformation S A (X) := X − AXA T for A ∈ R n×n that is positive semidefinite preserving (i.e., S A (S n +) ⊆ S n +) and show that such transformation is strictly monotone if and only if it is nondegenerate. We also show that a positive semidefinite pres...

In the setting of semidenite linear complementarity problems on , we focus on the Stein Transformation , and show that is (strictly) monotone if and only if (

In the setting of semidefinite linear complementarity problems on Sn, the implications strictmonotonicity⇒P2⇒GUS⇒P are known. Here, P and P2 properties for a linear transformation L:Sn→Sn are respectively defined by: X∈Sn,XL(X)=L(X)X⪯0⇒X=0 and X⪰0,Y⪰0, (X−Y)[L(X)−L(Y)](X+Y)⪯0⇒X=Y; GUS refers to the global unique solvability in semidefinite linear c...

In this paper, we present some new results for the semidefinite linear complementarity problem (SDLCP). In the first part, we introduce the concepts of (i) nondegeneracy for a linear transformation $L:{\cal S}^n \rightarrow {\cal S}^n$ and (ii) the locally-star-like property of a solution point of an SDLCP(L,Q) for $Q\in {\cal S}^n$, and we relate...

We correct an error in the statement of Theorem 8 in [1].

We correct an error in the statement of Theorem 8 in [1].

Given a linear transformation L:? n →? n and a matrix Q∈? n , where ? n is the space of all symmetric real n×n matrices, we consider the semidefinite linear complementarity problem SDLCP(L,? n +,Q) over the cone ? n + of symmetric n×n positive semidefinite matrices. For such problems, we introduce the P-property and its variants, Q- and GUS-prope...

For a Frechet differentiable function f from a closed rectangle Q in Rn into Rn, a result of Gale and Nikaido essentially asserts that f is a P-function on Q if the Jacobian matrix Jf(x) is a P-matrix for all x∈Q, and a result of More and Rheinboldt asserts that f is a P0-function on Q if and only if Jf(x) is a P0-matrix for all x∈Q. In this articl...

A nonempty set %plane1D;49E; in ℝn×n is said to have the row-P-property if every row representative of %plane1D;49E; is a P-matrix. We show that this property is equivalent to saying that for every nonzero x in ℝn there is an index i with x1(Mx)1 > 0 for all M∈%plane1D;49E;. We relate this concept to the unique solvability of certain nonlinear comp...