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

Abstract

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.