Paper ID sheet UCLINMA2008.030
 Title

Numerical representations of a universal subspace flow for linear programs
 Authors
 P.A. Absil
 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 $\mathrm{Gr}(m,n)$ as the set of all $n\times n$ projection
matrices of rank $m$, we view it as the set $\mathbb{R}_*^{m\times n}$ of all full column
rank $n\times m$ matrices, quotiented by the right action of the
general linear group $\mathrm{GL}(m)$. We propose a class of flows in $\mathbb{R}_*^{m\times n}$
that project to the flow on the Grassmannian. This approach requires
much less storage space when $n\gg m$ (i.e., there are many more
constraints than variables in the dual formulation).
One of the flows in $\mathbb{R}_*^{m\times n}$, that leaves invariant the set of
orthonormal matrices, turns out to be a particular version of a matrix
differential equation known as Oja's flow. We also point out that the
flow in the set of projection matrices admits a double bracket
expression.
 Key words
 Linear programming; Grassmannian; Grassmann manifold; Stiefel manifold; ordinary differential equation; Oja's flow; double bracket flow
 Status
 Communications in Information and Systems, Special Issue Dedicated to the 70th Birthday of Roger W. Brockett: Part I, Volume 8, Number 2, pp. 7184, 2008.
 Download

[Home]