How good are interior point methods? Klee–Minty cubes tighten iteration-complexity bounds

Abstract

By refining a variant of the Klee-Minty example that forces the central path to visit all the vertices of the Klee-Minty n-cube, we exhibit a nearly worst-case example for pathfollowing interior point methods. Namely, while the theoretical iteration-complexity upper bound is O(2 n n 5 2 ), we prove that solving this n-dimensional linear optimization problem requires at least 2 n − 1 iterations.

Central Path Curvature and Iteration-Complexity for Redundant Klee—Minty Cubes

“…In particular, we give a bound of O(n2 3n ) on the number of redundant constraints when the distances to those are chosen uniformly. When these distances are chosen to decay geometrically, we give a slightly tighter bound of the same order n 3 2 2n as in [4]. The behavior of the central path suggests that any feasible path-following interior-point method will take at least order 2 n iterations to solve this problem.…”

“…. , n, then (3) already implies (4). Therefore, it is left to show that d can be chosen such that h = ⌊ h⌋ satisfies the domination condition above.…”

“…Nesterov and Todd [9] studied the Riemannian curvature of the central path in particular relevant to the so-called short-step methods. We follow a constructive approach originated in [3,4] which is driven by the geometrical properties of the central path to address these questions.…”

Identity Matrix

“…In subsequent papers the number of redundant inequalities are reduced significantly. By decaying geometrically the distances of the redundant constraints to the corresponding facets, Deza, Nematollahi and Terlaky (2008) show that the number of the inequalities N can be reduced to Oðn 3 2 2n Þ and that after Oð ffiffiffi ffi N p nÞ iterations, a standard rounding procedure allows identification of the optimal solution. This results tighten the gap between iteration-complexity lower and upper bounds.…”

Smoothed analysis of condition numbers and complexity implications for linear programming

“…We thus partially close the gap between the theoretical worst-case bound and practical observations. Recently, Deza, Nematollahi and Terlaky [DET08] have proved a lower bound of Ω( n/ log 3 n) on the complexity of path-following interior point methods. This lower bound holds even when they only require the algorithm to produce a solution of small duality gap.…”