Lyapunov rank of polyhedral positive operators

IfK is a closed convex cone and if L is a linear operator having L (K) ⊆ K, then L is a positive operator on K and L preserves inequality with respect to K. The set of all positive operators on K is denoted by π (K). If K∗ is the dual of K, then its complementarity set is C (K) := {(x, s) ∈ K ×K | 〈x, s〉 = 0} . Such a set arises as optimality conditions in convex optimization, and a linear operator L is Lyapunov-like on K if 〈L (x), s〉 = 0 for all (x, s) ∈ C (K). Lyapunov-like operators help us find elements of C (K), and the more linearly-independent operators we can find, the better. The set of all Lyapunov-like operators on K forms a vector space and its dimension is denoted by β (K). The number β (K) is the Lyapunov rank of K, and it has been studied for many important cones. The set π (K) is itself a cone, and it is natural to ask if β (π (K)) can be computed, possibly in terms of β (K) itself. The problem appears difficult in general. We address the case where K is both proper and polyhedral, and show that β (π (K)) = β (K) in that case.