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

Let K⊆IRn denote the second-order cone. Given an n×nn×n real matrix M   and a vector q∈IRn, the second-order cone linear complementarity problem SOLCP(M,q)SOLCP(M,q) is to find a vector x∈IRn such thatx∈K,y:=Mx+q∈KandyTx=0.We say that M∈QM∈Q if SOLCP(M,q)SOLCP(M,q) has a solution for all q∈IRn. An n×nn×n real matrix A is said to be a Z-matrix with respect to KK iff:x∈K,y∈KandxTy=0  ⟹xTMy≤0.Let ΦM(q)ΦM(q) denote the set of all solutions to SOLCP(M,q)SOLCP(M,q). The following results are shown in this paper:
• If M∈Z∩QM∈Z∩Q, then ΦMΦM is Lipschitz continuous if and only if M   is positive definite on the boundary of KK.
• If M   is symmetric, then ΦMΦM is Lipschitz continuous if and only if M is positive definite.