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Lyapunov rank of polyhedral positive operators

Michael Orlitzky (michael***at***orlitzky.com)

Abstract: If K is a closed convex cone and if L is a linear operator having L(K) a subset of 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 pi(K). If J is the dual of K, then its complementarity set is C(K) := {(x,s) in (K,J) | = 0} . Such a set arises as optimality conditions in convex optimization, and a linear operator L is Lyapunov-like on K if = 0 for all (x,s) in 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 beta(K). The number beta(K) is the Lyapunov rank of K, and it has been studied for many important cones. The set pi(K) is itself a cone, and it is natural to ask if beta(pi(K)) can be computed, possibly in terms of beta(K) itself. The problem appears difficult in general. We address the case where K is both proper and polyhedral, and show that beta(pi(K)) is the square of beta(K) in that case.

Keywords: lyapunov rank, bilinearity rank, positive operators, completely-positive cone, copositive cone

Category 1: Linear, Cone and Semidefinite Programming

Category 2: Complementarity and Variational Inequalities

Category 3: Convex and Nonsmooth Optimization (Convex Optimization )

Citation: Linear and Multilinear Algebra (accepted)


Entry Submitted: 03/12/2017
Entry Accepted: 03/13/2017
Entry Last Modified: 05/15/2017

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