Lyapunov rank of polyhedral positive operators
Michael Orlitzky · Linear and Multilinear Algebra · 2017
If K is a closed convex cone and if L is a linear operator having , 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 . If is the dual of K, then its complementarity set isSuch a set arises as optimality conditions in convex optimization, and a linear operator L is Lyapunov-like on K if for all . 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 . The number is the Lyapunov rank of K, and it has been studied for many important cones. The set is itself a cone, and it is natural to ask if can be computed, possibly in terms of itself. The problem appears difficult in general. We address the case where K is both proper and polyhedral, and show that in that case.