Optimization Seminar
Location
Mathematics/Psychology : 401
Date & Time
September 20, 2018, 10:30 am – 12:00 pm
Description
Title: On subtangentiality and Z-operators
Speaker: Michael Orlitzky
Abstract:
Let K be a closed convex cone with dual cone dual(K) in a finite-dimensional real Hilbert space V. If L is a linear operator on V, then one definition says that L is a Z-operator on K if
<L(x), s> <= 0 for all x in K and s in dual(K) such that <x, s> = 0.
The Z-operators are generalizations of Z-matrices (whose off-diagonal elements are nonpositive) and they arise in dynamical systems, economics, game theory, and elsewhere. It is known that -L is exponentially-positive on K if and only if L is a Z-operator on K.
The outward normal cone to K at a point x on its boundary is
N_K(x) := {v in V | <v, y-x> <= 0 for all y in K},
and we say that z in V is subtangential to K at x if -z in dual(N_K(x)). It is also known that -L is exponentially-positive on K if and only if L(x) is subtangential to K at every point x on its boundary. The concept of subtangentiality is thus apparently connected to that of a Z-operator. We show that the connection can be made explicit without much trouble, thus providing an intuitive geometric interpretation for one modern definition of a Z-operator.
Speaker: Michael Orlitzky
Abstract:
Let K be a closed convex cone with dual cone dual(K) in a finite-dimensional real Hilbert space V. If L is a linear operator on V, then one definition says that L is a Z-operator on K if
<L(x), s> <= 0 for all x in K and s in dual(K) such that <x, s> = 0.
The Z-operators are generalizations of Z-matrices (whose off-diagonal elements are nonpositive) and they arise in dynamical systems, economics, game theory, and elsewhere. It is known that -L is exponentially-positive on K if and only if L is a Z-operator on K.
The outward normal cone to K at a point x on its boundary is
N_K(x) := {v in V | <v, y-x> <= 0 for all y in K},
and we say that z in V is subtangential to K at x if -z in dual(N_K(x)). It is also known that -L is exponentially-positive on K if and only if L(x) is subtangential to K at every point x on its boundary. The concept of subtangentiality is thus apparently connected to that of a Z-operator. We show that the connection can be made explicit without much trouble, thus providing an intuitive geometric interpretation for one modern definition of a Z-operator.
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