On the Lipschitzian property in linear complementarity problems over symmetric cones

Let V be a Euclidean Jordan algebra with symmetric cone K. We show that if a linear transformation L on V has the Lipschitzian property and the linear complementarity problem LCP(L,q) over K has a solution for every invertible q∈V, then 〈L(c),c〉>0 for all primitive idempotents c in V. We show that the converse holds for Lyapunov-like transformations, Stein transformations and quadratic representations. We also show that the Lipschitzian Q-property of the relaxation transformation RA on V implies that A is a P-matrix.