& \ldots  & 1 & 0 \\
2^m &  2^{m-1} & 2^{m-2} & \ldots &  2^2  & 1 \\
\end{array}
\right]
\left[ \begin{array}{c}
x_1 \\ x_2 \\ \vdots \\  \vdots \\ x_{m-1} \\ x_m
\end{array} \right] 
\le 
\left[ \begin{array}{c}
5 \\ 25 \\ \vdots \\  \vdots \\ 5^{m-1} \\ 5^m
\end{array} \right] 
  \\
& x_i \ge 0 \hspace{0.1in} i=1, \ldots, m.
\end{array}
\label{1stProblem}\end{aligned}$$ The optimizer is $[ 0, \ldots, 0,5^m ]$ with optimal objective function $-5^m$.

The second variant of Klee-Minty cube is given in [@ibrahima13]: $$\begin{aligned}
\begin{array}{cl}
\min & -\sum_{i=1}^m 10^{m-i} x_i \\
\mbox{subject to} & 
2 \sum_{j=1}^{i-1} 10^{i-j} x_j +x_i \le 100^{i-1}
 \hspace{0.1in} i=1, \ldots, m,  \\
& x_i \ge 0 \hspace{0.1in} i=1, \ldots, m.
\end{array}
\label{2ndProblem}\end{aligned}$$ The optimizer is $[ 0, \ldots, 0,10^{2(m-1)}]$ with optimal objective function $-10^{2(m-1)}$.

The third variant of Klee-Minty cube is given in [@km11] (its standard form was discussed in the previous section): $$\begin{aligned}
\begin{array}{cl}
\min & -\sum_{i=1}^m   x_i \\
\mbox{subject to} & 
x_1 \le 1, \\
& 2 \sum_{i=1}^{k-1} x_i +x_k \le 2^k-1
 \hspace{0.1in} k=2, \ldots, m, 
  \\
& x_i \ge 0 \hspace{0.1in} i=1, \ldots, m.
\end{array}
\label{3rdProblem}\end{aligned}$$ The optimizer is $[ 0, \ldots, 0,2^m-1]$ with optimal objective funtion $-(2^m-1)$. The test 