# Should I replace equality constraints by inequality constraints if the variables in the constraint appear in the objective?

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 $$x_i$$.