I have a simple model such as:
\begin{align}\max&\quad z=X_1+X_2+X_3+X_4\\\text{s.t.}&\quad y_1+y_2+y_3+y_4=2\\&\quad X_1 \leq y_1\\&\quad X_2 \leq y_1+y_2\\&\quad X_3 \leq y_2+y_3\\&\quad X_4 \leq y_1+y_4\\&\quad x,y \in \{0,1\}.\end{align}
The above formulation can be simplified by removing $X_1$ variable, $X_1 \leq y_1$ constraint and adding $y_1$ to the objective function to get:
\begin{align}\max&\quad z=y_1+X_2+X_3+X_4\\\text{s.t.}&\quad y_1+y_2+y_3+y_4=2\\&\quad X_2 \leq y_1+y_2\\&\quad X_3 \leq y_2+y_3\\&\quad X_4 \leq y_1+y_4\\&\quad x,y \in \{0,1\}\end{align}
because
- The direction of optimization is maximization,
- Objective function coefficients are $1$,
- $y$'s are bounding $x$'s and
- Variables are all binary.
Is it possible to assess any results on the polytopes or (extreme points of the polytopes) of these two formulations, something like "these two polytopes are equivalent as they yield the same optimal solution with this objective function"? What would be your approach to prove the equivalence of these two formulations?