m$.

-   The primal problem (\[LP\]) has an optimal solution.

-   Initial basic feasible solution $\x^0$ is given and is not an optimizer.

-   All basic feasible solutions are bounded above and below, more specifically, for all $i \in B \subset \mathcal{B}$, $ \delta \le x_i \le \gamma$.

The first three assumptions are standard. The last assumption implies that the primal problem (\[LP\]) is not degenerate. We also denote by $\x^*= (\x_{B^*}, \x_{N^*})$ the optimal basic solution of (\[LP\]) with $\x_{B^*}=\A_{B^*}^{-1} \b \ge \0$ and $\x_{N^*}=\0$, by $(\y^*, \s^*)$ the optimal basic solution of the dual problem (\[DP\]) with $\y^*=\A_{B^*}^{-\Tr} \c_{B^*}$, $\s_{B^*}=\0$, and $\s_{N^*}=\c_{N^*}-\A_{N^*}^{\Tr}\A_{B^*}^{-\Tr}\c_{B^*}$, by $z^* = \c^{\Tr} \x^* = \b^{\T}\y^*$ the optimal value. Using $B-N$ partition, we can write the primal problem as $$\begin{aligned}
\begin{array}{cl}
\min &  \c_B^{\T}\x_B + \c_N^{\T}\x_N,  \\
\mbox{\rm subject to} 
&  \A_B\x_B + \A_N\x_N=\b, 
\hspace{0.1in} \x_B \ge \0, \hspace{0.1in}  \x_N \ge \0.
\end{array}
\label{LP1}\end{aligned}$$ Since $\A_B$ is non-singular, we can rewrite (\[LP1\]) as $$\begin{aligned}
\begin{array}{cl}
\min & \c_B^{\T}\A_B^{-1} \b 
+ (\c_N - \A_N^{\T} \A_B^{-\Tr} \c_B)^{\Tr} \x_N, \\
\mbox{\rm subject to} 
& \x_B = \A_B^{-1} \b - \A_B^{-1}\A_N\x_N, 
\hspace{0.1in} \x_B \ge \0, \hspace{0.1in}  \x_N \ge \0.
\end{array}
\label{LP2}\end{aligned}$$ Let superscript $k$ represent the $k$th iteration, the 