Location
Mathematics/Psychology : 401
Date & Time
September 25, 2014, 10:30 am – 11:30 am
Description
Speaker: Michael Orlitzky
Title: Lyapunov Rank and Perfect Cones
Abstract:
Let K in R^n be a proper cone with dual K^*. We define the complementarity set of K to be
C(K) := { (x,s) : x in K, s in K^*, <x,s> = 0 }.
We say that a matrix/transformation L on R^n is Lyapunov-like if
<Lx,s> = 0 for all (x,s) in C(K).
The dimension of the linear space of all such transformations is called the Lyapunov rank of K and is denoted by beta(K). This number was studied by Rudolf et al. for its importance in solving certain optimization problems over K.
In the first part of the talk we show that beta(K) is bounded above by (n-1)^2, thus improving the previously known bound of n^2-n. The new bound is sharp for n=3.
A perfect cone is a cone whose complementarity set C(K) can be expressed in terms of n linearly-independent Lyapunov-like transformations. In the second part of the talk, we show that beta(K) greater than or equal to n is a necessary and sufficient condition for a cone being perfect.
Title: Lyapunov Rank and Perfect Cones
Abstract:
Let K in R^n be a proper cone with dual K^*. We define the complementarity set of K to be
C(K) := { (x,s) : x in K, s in K^*, <x,s> = 0 }.
We say that a matrix/transformation L on R^n is Lyapunov-like if
<Lx,s> = 0 for all (x,s) in C(K).
The dimension of the linear space of all such transformations is called the Lyapunov rank of K and is denoted by beta(K). This number was studied by Rudolf et al. for its importance in solving certain optimization problems over K.
In the first part of the talk we show that beta(K) is bounded above by (n-1)^2, thus improving the previously known bound of n^2-n. The new bound is sharp for n=3.
A perfect cone is a cone whose complementarity set C(K) can be expressed in terms of n linearly-independent Lyapunov-like transformations. In the second part of the talk, we show that beta(K) greater than or equal to n is a necessary and sufficient condition for a cone being perfect.
Tags: