Given a minimization problem with a cost function: $3x_1 + x_2 + \beta x_3 + \alpha x_4$
Under the constraints: $$2x_1 + \frac{1}{2}x_2 + 5x_3 + \frac{1}{2}x_4 = 7\\ 3x_1 + \frac{1}{2}x_2 +x_3 -\frac{1}{2}x_4 = \gamma\\ \overline x \geq 0$$
I want to find out what values of the parameters $\alpha, \beta, \gamma$ make it such that the basis ${2,4}$ is an optimal solution.
For the basis ${2,4}$ to be a feasible solution, I got that $-7 \leq \gamma \leq 7$, and then I tried going over all the possible neighbors of ${2,4}$ and checking to see which parameters cause the objective function to be non-increasing, but I got stuck.
I feel as if I'm missing something here, but I'm not sure what.
Any help would be welcome!
You have just to impose that the basis $$B=\begin{pmatrix} 1/2 & 1/2\\ 1/2 & -1/2\end{pmatrix}$$ is primal-feasible, that is $$B^{-1} b \geq 0$$ and dual-feasible, that is $$c^{\top} - c_B^{\top}B^{-1}A \geq 0$$