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. Copyright (C) by Springer-Verlag. Foundations of Computational Mathewmatics 2 (2002), 333–361. AMS Classification Numbers: 52A41, 53C22, 68Q25, 90C22, 90C25, 90C51, 90C60. CORE, Catholic University of Louvain, Louvain-Neuve, Belgium ([email protected]). This author was supported in part by the Mathematical Sciences Research Institute, UC Berkeley, USA. School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853, USA ([email protected]). This author was supported in part by NSF through grant DMS-9805602 and ONR through grant N00014-96-1-0050, and also by the Mathematical Sciences Research Institute, UC Berkeley, USA.