A simpler and tighter redundant Klee–Minty construction

Abstract

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.

Identity Matrix

“…The tightest result is given by Nematollahi and Terlaky (2008). The redundant constraints are placed parallel to the coordinate hyperplanes at geometrically decaying distances, so only N ¼ Oðn2 2n Þ redundant inequalities are needed to force the central path to follow the simplex path of the n-dimensional Zinchenko (2008, 2009), substantiating the relationships and presenting partial results.…”

“…While adding redundant constraints do not change the set of feasible solutions, the analytic center and the central path change. The redundant Klee-Minty example of Nematollahi and Terlaky (2008) is given as: min…”

Twenty-Five Years of Interior Point Methods

“…This is precisely the content of Proposition 1; i.e., it proves that the analytic center of C n is in the δ-neighborhoodĈ n δ of the initial vertex v {n} . The proof of the proposition can be found in Nematollahi and Terlaky [63]. Proposition 1.…”

“…In this chapter we discuss the tightest result to date. In Nematollahi and Terlaky [63], the authors simplify the construction given in Deza et al [27]. They are placing the redundant constraints parallel to the coordinate hyperplanes at geometrically decaying distances and show that only N = O(n2 2n ) redundant inequalities are needed to bend the central path along the edges of the n-dimensional Klee-Minty cube.…”

A redundant Klee–Minty construction with all the redundant constraints touching the feasible region

“…, 2 5 ) T , and the number of the redundant constraints is significantly reduced to N = O(n 3 2 2n ). A simplified construction, where the number of the redundant constraints is further reduced to N = O(n2 2n ), is presented in [9] by placing the redundant constraints parallel to the coordinate hyperplanes at geometrically decaying distances d…”