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\begin{document}
\section{}
(Reference: ???)
~~\\
~~\\
~~\\

Consider the LP problem
\beqn{eq:LP}
\begin{array}{cc}
\mbox{maximize}&c^Tx \\
\mbox{subject to}&Ax=b \\
 & x\geq 0,
\end{array}
\eeqn
where $c,x \in \mathbb{R}^n$, $b\in \mathbb{R}^m$, 
and $A$ is $m \times n$ matrix. 
The tableau with respect to the basis $B$ and nonbasis $N$ is:
\beqn{eq:LPtableau}
\begin{array}{ccccc}
&x_B &+ \bar{A}_Nx_N &=& \bar{b} \\
 z  &   &  -\bar{c}_Nx_N &=& \bar{v} 
\end{array}
\eeqn

Following is a description of criss-cross pivoting:
\begin{alg}
~~\\
\begin{enumerate}
\item
Let $r$ be the smallest index satisfying:
\[  r \in B, \bar{b}_r < 0  ~~\mbox{\underline{or}}~~
  r \in N, -\bar{c}_r < 0.
\]
If $r$ does not exist, then {\bf STOP}. The tableau is \underline{optimal}.
\item
Let $s$ be the smallest index satisfying: 
\[  r \in B, s\in N, \bar{a}_{rs} < 0  ~~\mbox{\underline{or}}~~
  r \in N, s \in B, \bar{a}_{sr} > 0.
\]
If $s$ does not exist, then {\bf STOP}. The tableau is \underline{infeasible}
if $r \in B$ and \underline{dual-infeasible} if $r \in N.$
\item
Where
\[ (\bar{r},\bar{s}) = \left\{ \begin{array}{cc}
    (r,s) & \mbox{if}~ r \in B \\
    (s,r) & \mbox{if}~ s \in B 
\end{array} 
  \right.
 \]
``pivot'' on $(\bar{r}, \bar{s}  ),$
That is, change the basis $B$ by:\\
replacing $r$ by $s$ if $r \in B$\\
\underline{or} replacing $s$ by $r$ if $r \in N$.
\end{enumerate}
\end{alg}

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