Suppose I am minimizing a MILP and my objective is simply $\sum_i{x_i}$, where $x_i$ is binary. I also have a constraint saying that $1-\sum_j y_j = x_i$, $\forall i \in \mathcal{S}$, $\forall j \in \mathcal{S}$ where $\mathcal{S}$ is the set defining the indices of my variables and $y_i$ is binary. I have many other constraints on $y_i$, but none other on $x_i$. Basically, $x_i$ is going to be 1 (and contribute to the cost function) whenever $\sum_j y_j$ is not 1. Note that I have other constraints that force $\sum_j y_j \leq 1$.
Does it make sense to replace $1-\sum_j y_j = x_i$ by $1-\sum_j y_j \leq x_i$? Where by "make sense" I mean would the solver solve the problem more quickly by replacing the equality constraint by an inequality one?
Of course, with the inequality constraint, $x_i$ can be either 1 or 0 even when $\sum_j y_j$ is 1. However, since $x_i$ appears in the cost function and I'm minimizing, the solver would push $x_i$ to 0 if $\sum_j y_j =1$, so we would still get the desired value for $x_i$.
