Klee–Minty’s LP and upper bounds for Dantzig’s simplex method

Abstract

get two upper bounds for the number of different basic feasible solutions generated by Dantzig's simplex method. The size of the bounds highly depends on the ratio between the maximum and minimum values of all the positive elements of basic feasible solutions. In this paper, we show some relations between the ratio and the number of iterations by using an example of LP, which is a simple variant of Klee-Minty's LP. We see that the ratio for the variant is equal to the number of iterations by Dantzig's simplex method for solving it. This implies that it is impossible to get a better upper bound than the ratio. We also give improved results of the upper bounds.

Efficient computation of a canonical form for a matrix with the generalized P-property

“…The sequence of papers by Kitihara and Mizuno, of which [8] is one example, generalizes Ye's results to other linear programs.…”

On the number of solutions generated by the dual simplex method

“…Kitahara and Mizuno [4] show that these bounds are almost tight in the sense that there are no upper bounds smaller than γ/δ.…”

Small Degenerate Simplices Can Be Bad for Simplex Methods

“…The linear optimization instance (LO 0 ) considered by Kitahara and Mizuno in [4,5], with x ∈ R m , is:…”

“…Thus, 2 m − 1 iterations may be required to solve the standard form of (LO 0 ) as observed by Kitahara and Mizuno [4,5].…”

“…Thus, the simplex method requires 2 m − 1 iterations. In this note, we essentially perturb the right-hand-side of a Klee-Minty cube considered by Kitahara and Mizuno [4,5] so that the feasible region becomes a full dimensional simplex. Further perturbing the right-hand-side, the feasible region is reduced to a zero-dimensional simplex, i.e.…”