An Analogue of the Klee-Walkup Result for Sonnevend's Curvature of the Central Path

For linear optimization (LO) problems, we consider a curvature integral first introduced by Sonnevend et al. (1991). Our main result states that in order to establish an upper bound for the total Sonnevend curvature of the central path, it is sufficient to consider only the case when n = 2m. This also implies that the worst cases of LO problems for path-following algorithms can be reconstructed for the case of n = 2m. As a by-product, our construction yields an asymptotically Ω(n) worst-case lower bound for Sonnevend’s curvature. Our research is motivated by the work of Deza et al. (2008) for the geometric curvature of the central path, which is analogous to the Klee-Walkup result for the diameter of a polytope.