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On the Lipschitz continuity of the solution map in linear complementarity problems over second-order cone

01-12-2016, Balaji Ramamurthy, 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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Positive definite and Gram tensor complementarity problems

01-05-2018, Balaji, R., 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.