An Analogue of the Klee–Walkup Result for Sonnevend’s Curvature of the Central Path

Abstract

For linear optimization problems, we consider a curvature integral of the central path which was first introduced by Sonnevend et al. (Math Progr 52:527–553, 1991). Our main result states that in order to establish an upper bound for the total Sonnevend curvature of the central path of a polytope, it is sufficient to consider only the case where the number of inequalities is twice as many as the number of variables. This also implies that the worst cases of linear optimization problems for certain path-following interior-point algorithms can be reconstructed for the aforementioned case. As a by-product, our construction yields an asymptotically worst-case lower bound in the order of magnitude of the number of inequalities for Sonnevend’s curvature. Our research is motivated by the work of Deza et al. (Electron Notes Discrete Math 31:221–225, 2008) for the geometric curvature of the central path, which is analogous to the Klee–Walkup result for the diameter of a polytope.