Show in the Simplex Method if a variable x_j leaves the basis, it cannot enter the basis in the next iteration.
First of all, since this is a homework question, I will assume it is true.
Intuitively, if I think about this, it would simply be moving back to the previous position, which would not increase the objective value. The Simplex Method only makes moves based on if the objective value increases.
Now, if I think about this in the Tableau method, the variable that leaves always has an objective value the opposite of what we want to increase (or decrease) the objective function, which would be why it does not enter the basis in the next iteration.
But, does anyone have an idea of how I can show this precisely?