before an optimizer is found. \[totalD\]

We may also partition $\x^k$ using $(B^*,N^*)$ as $$\x^k= \left[ \begin{array}{c}
\x_{B^*}^k \\ \x_{N^*}^k \end{array} \right].$$ This gives $$\begin{aligned}
\A_{B^*}\x_{B^*}^k +\A_{N^*}\x_{N^*}^k= \b, \nonumber\end{aligned}$$ and $$\begin{aligned}
\x_{B^*}^k  
= \A_{B^*}^{-1} \b -\A_{B^*}^{-1} \A_{N^*}\x_{N^*}^k.
\nonumber \end{aligned}$$ Similar to the derivation of (\[1stEst\]), we have $$\begin{aligned}
\c^{\Tr} \x^k  = \c^{\Tr} \x^* + 
({\c}_{N^*}- {\c}_{B^*}\A_{B^*}^{-1} \A_{N^*})^{\Tr} \x_{N^*}^k
= z^* + \bar{\c}_{N^*}^{\Tr}\x_{N^*}^k.
\nonumber \end{aligned}$$ If $x_{j}^k > 0$, we have $x_{j}^k \in B^k$. For $j \in {N^*} \cap B^k$, since $\bar{\c}_{N^*} \ge \0$, we have $$\begin{aligned}
z^* & \ge  \c^{\Tr} \x^k -   
\max \{ x_j^k   ~|~ j  \in N^* \cap B^k \}
\| \bar{\c}_{N^*}  \|_1.\end{aligned}$$

If $N^* \ne N^k$, i.e., $\x^k$ is not an optimizer, it must have $\bar{\c}_{N^*}^{\Tr}\x_{N^*}^k >0$. Therefore, there are $j^* \in N^*$ such that $$\bar{c}_{j^*} >0 \hspace{0.1in} \mbox{and}
\hspace{0.1in} x_{j^*}^k>0.
\label{NkBstar2}$$ This means that for $j^* \in N^* \cap B^k$, $0 < x_{j^*}^k \in B^k$ should be the leaving variable. The problem is that one does not know $N^*$ before an optimizer is found. \[totalDiff\]

Let $\gamma_{\ell}=\min_{k} x_{\jmath_2^k}$, where $x_{\jmath_2^k}$ is defined in (\[iniVar2\]), i.e., $x_{\jmath_2^k}$ is the longest step among all possible entering variables with $\bar{c}_{j^k} <0$ and $j^k \in N^k$; and define $$\delta_D =\min_{k} 