I have a 0-1 integer linear program for a set of $2n$ variables $S = \{x_1, ..., x_n, y_1, ..., y_n\}$. My objective is to maximize the number of pairs $(x_i, y_i)$ such that $x_i = y_i$, $i = 1, ..., n$.
$$ \max \sum_{i=1}^n \mathbb{1}(x_i = y_i) $$
where $\mathbb{1}(x_i = y_i) = 1$ if and only if $x_i = y_i$, $i = 1, ..., n$.
The constraints are not important in this context. You may think the only constraints are: $$x_i \in \{0, 1\}, y_i \in \{0, 1\}, i = 1, ..., n$$
Obviously, this objective is not linear w.r.t the variables. I am wondering if there is a workaround (e.g., additional constraints) to construct a different objective function that is (1) linear and (2) also maximizes the number of such pairs of variables.
