 
Summary: 02 Jun 2008 Tech. Report UCLINMA2008.030v1
Numerical representations of a universal subspace flow for linear
programs
P.A. Absil
June 2, 2008
This paper is dedicated to Roger Brockett on the occasion of his 70th birthday
Abstract
In 1991, Sonnevend, Stoer, and Zhao [Math. Programming 52 (1991) 527553] have
shown that the central paths of strictly feasible instances of linear programs generate
curves on the Grassmannian that satisfy a universal ordinary differential equation. Instead
of viewing the Grassmannian Gr(m, n) as the set of all n × n projection matrices of rank
m, we view it as the set Rn×m
of all full column rank n × m matrices, quotiented by
the right action of the general linear group GL(m). We propose a class of flows in Rn×m
that project to the flow on the Grassmannian. This approach requires much less storage
space when n m (i.e., there are many more constraints than variables in the dual
formulation). One of the flows in Rn×m
, that leaves invariant the set of orthonormal
matrices, turns out to be a particular version of a matrix differential equation known as
