There are two Mixed Integer Linear Programs. They have the same set of linear constraints constraints, but different objectives with variables $\mathbf{z}$ and $\mathbf{x}$.
The first objective is:
$\text{max}_{~\mathbf{z},\mathbf{x}} ~~~ \sum_{u=1}^{N_1} z_u$
where $z_i \in \{0,1\}$.
The second objective is:
$\text{max}_{~\mathbf{z},\mathbf{x}} ~~~ \sum_{u=1}^{N_1} z_u -\sum_{i=1}^{N_2} \sum_{j=1}^{N_3} \frac{1}{c_{i,j}}x_{i,j}$
where $z_i \in \{0,1\}$, $x_{i,j} \in \{0,1\}$, and values $c_{i,j}$ are given.
I came up with the second objective because I wanted to achieve the same optimal $\mathbf{z}^*$ in both programs, but to make optimal $\mathbf{x}^*$ sparser in the second program. I am sure that my approach in the second objective maximizes the number of non-zero elements in $\mathbf{z}$ and at the same time it minimizes the number of non-zero elements in $\mathbf{x}$. This happens due to the set of constraints that I have.
Question: Is there a standard mathematical tool/approach that I can use to prove my claim? I need to show that $\mathbf{z}^*$ is the same in both programs, and that $\mathbf{x}^*$ is sparser with the second objective, which are the things I wanted to achieve.
If you can show that, in the second problem, $\sum_{i=1}^{N_2} \sum_{j=1}^{N_3}\frac{1}{c_{i,j}} x_{i,j} < 1$ for any possible (optimal) $x$, then I think you can show that $\sum_i z_i$ is the same for optimal solutions of the two problems. I'm not sure how you find a more general proof.