The Lyapunov rank of an improper cone. (2nd January 2017) - Orlitzky, Michael
Abstract : Let K be a closed convex cone with dual K ∗ in a finite-dimensional real inner-product space  V . The complementarity set of K is C ( K ) = { ( x, s ) ∈ K × K ∗ | ⟨ x, s ⟩ = 0 } . We say that a linear transformation L : V → V is Lyapunov-like on K if ⟨ L ( x ), s ⟩ = 0 for   all   ( x, s ) ∈ C ( K ) . The dimension of the space of all such transformations is called the Lyapunov rank of  K . This number was introduced and studied by Rudolf et al.  [Bilinear optimality constraints for the cone of positive polynomials, Math. Program., Ser. B 129 (2011), pp. 5–31] for proper cones because of its connection to conic programming and complementarity problems. The assumption that K is proper turns out to be nonessential. We first develop the basic theory for cones that are merely closed and convex. We then devise a way to compute the Lyapunov rank of any closed convex cone and show that the Lyapunov-like transformations on a closed convex cone are related to the Lie algebra of its automorphism group. Next, we extend some results for proper polyhedral cones. Finally, we devise algorithms to compute both the space of all Lyapunov-like transformations and the Lyapunov rank of a polyhedral closed convex cone.