NO - Summary of Chapter 3 (Part V)

Given the basis/non-basis formulation of an LP: (dropping overlines cdots)

$$\begin{array}{rl}& \min c_B^t{B}^{-1}\left(b-N{x}_N\right)+c_N{x}_N\\ & x_N\ge 0\\ & x_N\ge 0\end{array}$$

We can say that, if $x$ is a vertex, then:

$$\begin{array}{rl}& x_B=B^{-1}b\\ & x_N=0\end{array}$$

Also one of the constraints enforces that:

$$x_N\ge 0$$

If we increase $x_N$ we have a change in the whole $A$ matrix, such that $B$ and $N$ exchange variables, so with these changes we can re-calculate $x_B=B^{-1}b-B^{-1}N{x}_N$

NOTE: $x_B$ can increase or decrease but we need to have $x_B\ge 0$ (constraint)

And find a new vertex of the problem, which might be a better solution.