Numerical Representations of a Universal Subspace Flow for Linear Programs

Abstract

In 1991, Sonnevend, Stoer, and Zhao [Math. Programming 52 (1991) 527--553] 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 $Grassmn$ as the set of all $n imes n$ projection matrices of rank $m$, we view it as the set $STmn$ of all full column rank $n imes m$ matrices, quotiented by the right action of the general linear group $GL(m)$. We propose a class of flows in $STmn$ that project to the flow on the Grassmannian. This approach requires much less storage space when $ngg m$ (i.e., there are many more constraints than variables in the dual formulation). One of the flows in $STmn$, 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.