On calculating analytic centers

Abstract

The analytic center of a polytope can be calculated in polynomial time by Newton's method. This note was motivated by papers of Renegar and Shub(88) and by Ye(89). We apply Smale's(86) estimates at one point for Newton's method to the problem of finding the analytic center of a polytope. The method converges globally in the appropriate norm. The ideas are then applied to obtain a possible benchmark for path following methods. When Smale's method is tractable its power stems not only from the fact that the information is concentrated at one point. There are 2 norms to estimate, not 3 as in the Kantorovich estimate. Moreover no estimate of the inverse of the derivative operator by itself is needed. The need for the norm of the inverse by itself often makes for coarse estimates. (kr)