Simultaneous Data Perturbations and Analytic Center Convergence

Abstract

The central path is an infinitely smooth parameterization of the nonnegative real line, and its convergence properties have been investigated since the mid 1980s. However, the central "path" followed by an infeasible-interior-point method relies on three parameters instead of one, and hence is a surface instead of a path. The additional parameters are included to allow for simultaneous perturbations in the cost vectors and right-hand side vectors. This paper provides a detailed analysis of the perturbed central path that is followed by infeasible-interior-point methods, and we characterize when such a path converges. We develop a set (Hausdorff) convergence property and show that the central paths impose an equivalence relation on the set of admissible cost vectors. We conclude with a technique to test for convergence under arbitrary, simultaneous data perturbations.