Newton Flow and interior Point Methods in Linear Programming

In this paper we take up once again the subject of the geometry of the central paths of linear programming theory. We study the boundary behavior of these paths as in Meggido and Shub [5], but from a different perspective and with a different emphasis. Our main goal will be to give a global picture of the central paths even for degenerate problems as solution curves of the Newton vector field, N(x), of the logarithmic barrier function which we describe below. See also Bayer and Lagarias [1], [2], [3]. The Newton vector field extends to the boundary of the polytope. It has the properties that it is tangent to the boundary on the boundary and restricted to any face of dimension i it has a unique source with unstable manifold dimension equal to i, the rest of the orbits tending to the boundary of the face. Every orbit tends either to a vertex or one of these sources in one of the faces. See the Corollary 4.1. This highly cellular structure of the flow lends itself to the conjecture that the total curvature of these central paths may be linearly bounded by the dimension n of the polytope. The orbits may be relatively straight, except for orbits which come close to an orbit in a face of dimension i which itself comes close to a singularity in a boundary face of dimension less than i. This orbit then is forced to turn to be almost parallel to the lower dimensional face so its tangent vector may be forced to turn as well. See the two figures at the end of this paper. As this process involves a reduction of the dimension of the face it can only happen the dimension of the polytopetimes. So our optimistic conjecture is that the total curvature of a central path is O(n). We have verified the conjecture in an average sense in Dedieu, Malajovich and Shub [4]. It is not difficult to give an example showing that O(n) is the best