NO - Summary of Chapter 3 (Part IV)

After we define $\left[{\overline{c}}_B,{\overline{c}}_N\right]$ in the same way we defined $\overline{x}=\left[{\overline{x}}_B,{\overline{x}}_N\right]$, we can write the LP:

$$\begin{array}{rl}& \min \overline{c}{}^t\cdot \overline{x}\\ & \overline{\overline{A}}\cdot \overline{x}=\overline{b}\\ & \overline{x}\ge 0\end{array}$$

as:

$$\begin{array}{rl}& \min {\overline{c}}_B{}^t\cdot {\overline{x}}_B+{\overline{c}}_N{}^t\cdot {\overline{x}}_N\\ & \overline{\overline{B}}\cdot {\overline{x}}_B+\overline{\overline{N}}\cdot {\overline{x}}_N=\overline{b}\\ & {\overline{x}}_B\ge 0\\ & {\overline{x}}_N\ge 0\end{array}$$

From the equation we obtain that:

$${\overline{x}}_B={\overline{\overline{B}}}^{-1}\cdot \left(\overline{b}-\overline{\overline{N}}\cdot {\overline{x}}_N\right)$$

So the LP becomes:

$$\begin{array}{rl}& \min {\overline{c}}_B{}^t\cdot {\overline{\overline{B}}}^{-1}\cdot \left(\overline{b}-\overline{\overline{N}}\cdot {\overline{x}}_N\right)+{\overline{c}}_N\cdot {\overline{x}}_N\\ & {\overline{x}}_B\ge 0\\ & {\overline{x}}_N\ge 0\end{array}$$