Primal-Dual Interior-Point Methods for Self-Scaled Cones

In this paper we continue the development of a theoretical foundation for efficient primal-dual interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled (see [NT97]). The class of problems under consideration includes linear programming, semidefinite programming and convex quadratically constrained quadratic programming problems. For such problems we introduce a new definition of affine-scaling and centering directions. We present efficiency estimates for several symmetric primal-dual methods that can loosely be classified as path-following methods. Because of the special properties of these cones and barriers, two of our algorithms can take steps that go typically a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier. 1. Introduction. This paper continues the development begun in [NT97] of a theoretical foundation for efficient interior-point methods for problems that are extensions of linear programming. Here we concentrate on symmetric primal-dual algorithms that can loosely be classified as path-following methods. While standard form linear programming problems minimize a linear function of a vector of variables subject to linear equality constraints and the requirement that the vector belong to the nonnegative orthant in n-dimensional Euclidean space, here this cone is replaced by a possibly non-polyhedral convex cone. Note that any convex programming problem can be expressed in this conical form. Nesterov and Nemirovskii [NN94] have investigated the essential ingredients necessary to extend several classes of interior-point algorithms for linear programming (inspired by Karmarkar's famous projective-scaling method [Ka84]) to nonlinear settings. The key element is that of a self-concordant barrier for the convex feasible region. This is a smooth convex function defined on the interior of the set, tending to +∞ as the boundary is approached, that together with its derivatives satisfies certain Lipschitz continuity properties. The barrier enters directly into functions used in path-following and potential-reduction methods, but, perhaps as importantly, its Hessian at any point defines a local norm whose unit ball, centered at that point, lies completely within the feasible region. Moreover, the Hessian varies in a well-controlled way in the interior of this ball. In [NT97] Nesterov and Todd introduced a special class of self-scaled convex cones and associated barriers. While they are required to satisfy certain apparently restrictive conditions, this class includes some important instances, for example the cone …