Representing the space of linear programs as a Grassmannian

We represent the space of linear programs as the space of projection matrices. Projection matrices of the same dimension and rank comprise a Grassmannian, which has rich geometric and algebraic structures. An ordinary differential equation on the space of projection matrices defines a path for each projection matrix associated with a linear programming instance and the path leads to a projection matrix associated with an optimal basis of the instance. In this way, any point (projection matrix) in the Grassmannian is connected to a stationary point of the differential equation. We will present some basic properties of the stationary points, in particular, the characteristics of eigenvalues and eigenvectors. We will show that there are only a finite number of stable points. Thus, the Grassmannian can be partitioned into a finite number of attraction regions, each associated with a stable point. The structures of the attraction regions will be important for applications which will be discussed at the end of this paper.


Department of Mathematics, National University of Singapore. March 2006.



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