On the complex complementarity problem

The complex linear complementarity problem considered here is the following: Find z such thatwhere S is a polyhedral convex cone in Cp, S* the polar cone, M ∈ Cp×p and q ∈ Cp.Generalizing earlier results in real and complex space, it is shown that if M satisfies RezHMz ≥ 0 for all z ∈ Cp and if the set satisfying Mz + q ∈ S*, z ∈ S is not empty, then a solution to the complex linear complementarity problem exists. If RezHMz > 0 unless z = 0, then a solution to this problem always exists.