# Equality constraints in mixed-integer optimization

Suppose I have a linear mixed-integer optimization problem of the form

$$MIP: min_{(x,y) \in \mathbb{R}^n \times \mathbb{Z}^m} c^\top x + d^\top y \hspace{0.2cm} \text{s.t.} \hspace{0.1cm} Ax+By \leq b, Cx+Dy =\gamma,$$ with $c\in \mathbb{R}^n$, $d \in \mathbb{R}^m$, $b \in \mathbb{R}^p$, $\gamma \in \mathbb{R}^q$, some $(p,n)$-matrix $A$, $(p,m)$-matrix $B$, $(q,n)$-matrix $C$ and $(q,m)$-matrix $D$.

Would a typical solver like GUROBI/ CPLEX apply some branch-and-cut alrorithm to solve this problem, or is there some "special treatment" for the equality constraints $Cx+Dy =\gamma$? In the literature, I can hardly find equality constraints in the problem formulation of $MIP$. Can someone tell me the reason for this?

I don't think there is any reason to use a special treatment for those kind of constraints. Any MIP solver can work perfectly fine with both inequality and equality constraints. There is no need to put equality constraints explicitly in text books, since you can replace them by two inequality constraints: $Cx+Dy\leq \gamma$ and $-Cx-Dy\leq -\gamma$.