Isotone solutions of parametric linear complementarity problems

The parametric linear complementarity problem is given by the conditions:q + αp + Mz ≥ 0, α ≥ 0, z ≥ 0, z^T(q + αp + Mz) = 0. Under the assumption that M is a P-matrix, Cottle proved that the solution map z(α) of the above problem is montonically nondecreasing in the parameter α for every nonnegative q and every p if and only if M is a Minkowski matrix. This paper examines whether a similar result holds in various other settings including a nonlinear case.