On the Riemannian Geometry Defined by Self-Concordant Barriers and Interior-Point Methods

We consider the Riemannian geometry defined on a convex set by the Hessian of a selfconcordant barrier function, and its associated geodesic curves. These provide guidance for the construction of efficient interior-point methods for optimizing a linear function over the intersection of the set with an affine manifold. We show that algorithms that follow the primal-dual central path are in some sense close to optimal. The same is true for methods that follow the shifted primal-dual central path among certain infeasible-interior-point methods. We also compute the geodesics in several simple sets.

• Type: Journal
• Year: 2002
• Where: FOCM
• Authors: Yu. E. Nesterov, Michael J. Todd
• Keywords: Associated Geodesic Curves, Efficient Interior-point Methods, FOCM 2002, Primal-dual Central Path