On the relationship between the curvature integral and the complexity of path-following methods in linear programming

Abstract

In this paper we study the complexity of primal-dual path-following methods which are a particular kind of interior point method for solving linear programs. In particular we establish a relationship between the complexity and the integral of a weighted curvature along the central trajectory (path) defined in the interior of the feasible solution space of a linear program. An important property of the trajectory, viz., that its higher order derivatives are bounded by a geometric series, is presented. Applying this property we show that the complexity of such methods is bounded from below and from above by the curvature integral. This theorem reduces the complexity analysis to a relatively simple problem of estimating the curvature integral.