basic matrix is lower triangle and it is easy to see the claim), this shows that $\delta_D = 1$ because either $x_1=1$ or $x_{m+1}=1$. In the first iteration, noticing that $B^0=\{ m+1, m+2,\ldots, 2m \}$ and the entering variable $\bar{x}_{\jmath_2^k}=x_m=2^m-1$. In view of Theorem \[mainThe\], it needs only one iteration to find the optimal solution and $\gamma_{\ell}=2^m-1$. This claim is verified in the numerical test in the next section for several variants of Klee-Minty cube.

Numerical test
==============

Klee-Minty cube and its variants have been used to prove that several popular simplex algorithms need exponential number of iterations in the worst case to find an optimizer. In this section, three variants of Klee-Minty cube [@greenberg97; @ibrahima13; @km11] are used to test the proposed algorithm.

The first variant of Klee-Minty cube is given in [@greenberg97]: $$\begin{aligned}
\begin{array}{cl}
\min & -\sum_{i=1}^m 2^{m-i} x_i \\
\mbox{subject to} & 
\left[ 
\begin{array}{cccccc}
1 & 0 & 0 & \ldots & 0 & 0 \\
2^2 & 1 & 0 & \ldots & 0 & 0 \\
2^3 & 2^2  & 1 & \ldots & 0 & 0 \\
\vdots &  \vdots & \vdots &  \ddots & 0 & 0 \\
2^{m-1} &  2^{m-2} & 2^{m-3} 