Q: Given the Linear Program (LP):
$\\$ Let $\beta = -1$ and $\gamma = -1$ be fixed. Find the range of $\alpha$ such that $\mathbf{x} = (2,0,-1)$ is an optimal solution.
My approach:
Since solving this problem graphically will be challenging, I am trying other means such as showing that $\bar c \geq \mathbf 0$ for a range of $\alpha$. However, I tried utilizing the formula $\bar c = c - c_{B}^{T} A_{B}^{-1}A$, but I am unsure of which basis $B$ to choose. If I let $B=\{1,2,3\}$, I will simply get $\bar c = \mathbf 0$, which doesn't seem quite right.. Some help and clarification will be deeply appreciated as I am rather lost in this area of linear optimization..