Abstract: The simplex and primal-dual interior point methods are currently the most computationally successful algorithms for linear optimization. While the simplex methods follow an edge path, the interior point methods follow the central path. Within this framework, the curvature of a polytope, defined as the largest possible total curvature of the associated central path, can be regarded as the continuous analogue of its diameter. In this talk we highlight links between the edge and central paths, and between the diameter and the curvature of a polytope. We recall continuous results of Dedieu-Malajovich-Shub, and discrete results of Holt-Klee and Klee-Walkup, as well as related conjectures such as the Hirsch conjecture which was disproved by Santos in 2012. We also present analogous results dealing with average and worst-case behaviour of the curvature and diameter of polytopes, including a recent result of Allamigeon, Benchimol, Gaubert, and Joswig who constructed a counterexample to the continuous analogue of the polynomial Hirsch conjecture.
Based on joint work with
Tamas Terlaky (Lehigh),
Feng Xie (Microsoft),
Yuriy Zinchenko (Calgary).