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.
By introducing redundant Klee-Minty examples, we have previously shown that the central path can be bent along the edges of the Klee-Minty cubes, thus having 2 n − 2 sharp turns in dimension n. In those constructions the redundant hyperplanes were placed parallel with the facets active at the optimal solution. In this paper we present a simpler and more powerful construction, where the redundant constraints are parallel with the coordinate-planes. An important consequence of this new construction is that one of the sets of redundant hyperplanes is touching the feasible region, and N , the total number of the redundant hyperplanes is reduced by a factor of n 2 , further tightening the gap between iteration-complexity upper and lower bounds.
By introducing some redundant Klee-Minty constructions, we have previously shown that the central path may visit every vertex of the Klee-Minty cube having 2 n − 2 "sharp" turns in dimension n. In all of the previous constructions, the maximum of the distances of the redundant constraints to the corresponding facets is an exponential number of the dimension n, and those distances are decaying geometrically. In this paper, we provide a new construction in which all of the distances are set to zero, i.e., all of the redundant constraints touch the feasible region.
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