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Summary: 02 Jun 2008 Tech. Report UCL-INMA-2008.030v1
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
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