Doctoral Dissertation Defense: Michael Orlitzky
Advisor: Dr. Muddappa Gowda
Friday, April 7, 2017 · 10 AM - 12 PM
Title: Positive operators, Z-operators, Lyapunov rank, and linear games over closed convex cones
Abstract:
Given a closed convex cone K with dual K^* in a finite-dimensional real Hilbert space H, a linear operator L is positive on K if L(K) is a subset of K, and a Z-operator on K if <L(x),s> is nonpositive for all x in K and s in H with <x,s> = 0. If both L and -L are Z-operators on K, then L is Lyapunov-like on K. These concepts generalize (respectively) the nonnegative, Z, and diagonal matrices. They appear in various fields including dynamical systems, optimization, economics, and game theory. Our contribution is to extend results for proper cones to general closed convex cones.
We extend a result of Tam describing the dual of the cone of positive operators. We compute its largest linear subspace, and then use the exponential map to connect positive and Z-operators. In particular, we show that both families share the same dimension and polyhedrality.
Motivated by optimization considerations, the Lyapunov rank of K is defined as the dimension of the space of all Lyapunov-like operators on K. We extend this concept from proper to closed convex cones, and provide an algorithm to efficiently compute the Lyapunov rank in that case. We further show that the Lyapunov rank of K is the dimension of the Lie algebra of the automorphism group of K in the general setting. We then improve an existing upper bound for the Lyapunov rank.
We introduce linear games on proper cones, and extend some results that Gowda and Ravindran formulated for self-dual cones. The more-general setting allows us to place linear games in the framework of conic programming.
Abstract:
Given a closed convex cone K with dual K^* in a finite-dimensional real Hilbert space H, a linear operator L is positive on K if L(K) is a subset of K, and a Z-operator on K if <L(x),s> is nonpositive for all x in K and s in H with <x,s> = 0. If both L and -L are Z-operators on K, then L is Lyapunov-like on K. These concepts generalize (respectively) the nonnegative, Z, and diagonal matrices. They appear in various fields including dynamical systems, optimization, economics, and game theory. Our contribution is to extend results for proper cones to general closed convex cones.
We extend a result of Tam describing the dual of the cone of positive operators. We compute its largest linear subspace, and then use the exponential map to connect positive and Z-operators. In particular, we show that both families share the same dimension and polyhedrality.
Motivated by optimization considerations, the Lyapunov rank of K is defined as the dimension of the space of all Lyapunov-like operators on K. We extend this concept from proper to closed convex cones, and provide an algorithm to efficiently compute the Lyapunov rank in that case. We further show that the Lyapunov rank of K is the dimension of the Lie algebra of the automorphism group of K in the general setting. We then improve an existing upper bound for the Lyapunov rank.
We introduce linear games on proper cones, and extend some results that Gowda and Ravindran formulated for self-dual cones. The more-general setting allows us to place linear games in the framework of conic programming.
