Suppose we have variables $x_1,x_2,y \in \{0,1\}$ such that $y=1$ if and only if $x_1 = x_2 = 1$ and we want maximize the value of $y$. I know that this reduces to the following Integer programming problem: \begin{align*} \max y&\\ x_1,x_2, y \in \{0,1\}\\ y \leq x_1\\ y\leq x_2.\\ \end{align*}
The linear relaxation of this problem is then given by: \begin{align*} \max y&\\ 0\leq x_1,x_2, y \leq 1\\ y \leq x_1\\ y\leq x_2.\\ \end{align*} Now in the relaxation we will simply get $y=\min\{x_1,x_2\}$. I was wondering whether there is some alternative formulation for the Integer Program such that, when we take the linear relaxation of the problem, the solution $y$ is stricly smaller than $\min\{x_1,x_2\}$. See also here for more information about reformulations.
This reformulation relies on the optimality condition that $y$ is maximized. The constraints enforce only $y\le \min(x_1,x_2)$, and $y< \min(x_1,x_2)$ is feasible but not optimal. If your problem has a different objective, you can instead impose an additional linear constraint $y \ge x_1 + x_2 - 1$ to enforce $(x_1=1 \land x_2=1) \implies y=1$, as described here.