Linear complementarity results for Z-matrices on Lorentz cone

Let K⊆RnK⊆Rn be the n  -dimensional Lorentz cone. Given an n×nn×n matrix M   and q∈Rnq∈Rn, the Lorentz-cone linear complementarity problem LCLCP(M,q)LCLCP(M,q) is to find an x∈Rnx∈Rn that satisfiesx∈K,y:=Mx+q∈KandyTx=0. We show that if M is a Z-matrix with respect to KK, then M   is positive stable if and only if LCLCP(M,q)LCLCP(M,q) has a non-empty finite solution set for all q∈Rnq∈Rn.