Now showing 1 - 6 of 6
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    A uniqueness result for linear complementarity problems over the Jordan spin algebra
    (15-08-2015)
    Given a Euclidean Jordan algebra (V,o,〈.,.〉) with the (corresponding) symmetric cone K, a linear transformation L:V→V and qεV, the linear complementarity problem LCP(L,q) is to find a vector xεV such thatxεK, y:=L(x)+qεK and xy=0. To investigate the global uniqueness of solutions in the setting of Euclidean Jordan algebras, the P-property and its variants of a linear transformation were introduced in Gowda et al. (2004) [3] and it is shown that if LCP(L,q) has a unique solution for all qεV, then L has the P-property but the converse is not true in general. In the present paper, when (V,o,〈,〉) is the Jordan spin algebra, we show that LCP(L,q) has a unique solution for all qεV if and only if L has the P-property and L is positive semidefinite on the boundary of K.
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    On the Lipschitz continuity of the solution map in linear complementarity problems over second-order cone
    (01-12-2016) ;
    Palpandi, K.
    Let K⊆IRn denote the second-order cone. Given an n×n real matrix M and a vector q∈IRn, the second-order cone linear complementarity problem SOLCP(M,q) is to find a vector x∈IRn such thatx∈K,y:=Mx+q∈KandyTx=0. We say that M∈Q if SOLCP(M,q) has a solution for all q∈IRn. An n×n real matrix A is said to be a Z-matrix with respect to K iff:x∈K,y∈KandxTy=0  ⟹xTMy≤0. Let ΦM(q) denote the set of all solutions to SOLCP(M,q). The following results are shown in this paper: • If M∈Z∩Q, then ΦM is Lipschitz continuous if and only if M is positive definite on the boundary of K.• If M is symmetric, then ΦM is Lipschitz continuous if and only if M is positive definite.
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    Linear complementarity results for Z-matrices on Lorentz cone
    (01-09-2015)
    Let K ⊃ Rn be the n-dimensional Lorentz cone. Given an n×n matrix M and q ε Rn, the Lorentz-cone linear complementarity problem LCLCP(M,q) is to find an x εRn that satisfiesx ε K,y:=Mx+q ε KandyTx=0. We show that if M is a Z-matrix with respect to K, then M is positive stable if and only if LCLCP(M,q) has a non-empty finite solution set for all q ε Rn.
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    Characterization of p-property for some z-transformations on positive semidefinite cone
    (01-01-2011)
    The P-property of the following two Z-transformations with respect to the positive semidefinite cone is characterized: (i) I-S, where S: Sn×n→Sn×n is a nilpotent linear transformation, (ii) I-L-1 A, where LA is the Lyapunov transformation defined on Sn×n by LA(X) = AX + XAT. (Here Sn×n denotes the space of all symmetric n×n matrices and I is the identity transformation.).
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    Simple expressions for the long walk distance
    (01-01-2013)
    Chebotarev, Pavel
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    Bapat, R. B.
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    The walk distances in graphs are defined as the result of appropriate transformations of the ∑k=0∞(tA)k proximity measures, where A is the weighted adjacency matrix of a connected weighted graph and t is a sufficiently small positive parameter. The walk distances are graph-geodetic, moreover, they converge to the shortest path distance and to the so-called long walk distance as the parameter t approaches its limiting values. In this paper, simple expressions for the long walk distance are obtained. They involve the generalized inverse, minors, and inverses of submatrices of the symmetric irreducible singular M-matrix L=ρI-A, where ρ is the Perron root of A. © 2013 Elsevier Inc. All rights reserved.
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    Positive definite and Gram tensor complementarity problems
    (01-05-2018) ;
    Palpandi, K.
    Given a continuous function [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.], the non-linear complementarity problem NCP(g,q) is to find a vector [InlineEquation not available: see fulltext.] such that x≥0,y:=g(x)+q≥0andxTy=0.We say that g has the Globally Uniquely Solvable (GUS)-property if NCP(g,q) has a unique solution for all [InlineEquation not available: see fulltext.] and C-property if NCP (g, q) has a convex solution set for all [InlineEquation not available: see fulltext.]. In this paper, we find a class of non-linear functions that have the GUS-property and C-property. These functions are constructed by some special tensors which are positive semidefinite. We call these tensors as Gram tensors.