Numerical Representations of a Universal Subspace Flow for Linear Programs∗

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 Gr(m, n) as the set of all n×n projection matrices of rank m, we view it as the set R ∗ 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 R ∗ 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 R ∗ , 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.