This problem is a Mixed Integer Quadratic Program that has a diagonal quadratic matrix of 1s. $$ \max_{x\text{ s.t. }Ax\leq b}\left\{ \ a'x-\frac{1}{2}x' \begin{bmatrix} 1 & 0 & \dots &0 \\ 0 & 1 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 &\dots & 1 \end{bmatrix}x\right\} $$ with the additional condition that $x_i\in\{0,\ 1\}$ for some $i$s.
Question: is there a way to rewrite this problem (or at least approximate its solution) as a Mixed Integer Linear Program?
Your objective is to maximize $$\sum_i a_i x_i - \frac{1}{2} \sum_i x_i^2.$$ If $x_i\in\{0,1\}$, you can replace $x_i^2$ with $x_i$. For the other $x_i$, you have $x_i^2 \ge 0$, so replacing the objective function with the linear function $$\sum_{i\in N} a_i x_i - \frac{1}{2} \sum_{i \in I} x_i,$$ where $I \subseteq N$ is the index set for the binary variables, yields an upper bound on the original objective because $$ \sum_{i\in N} a_i x_i - \frac{1}{2} \sum_{i \in I} x_i =\sum_{i\in N} a_i x_i - \frac{1}{2} \sum_{i \in I} x_i^2 \ge \sum_{i\in N} a_i x_i - \frac{1}{2} \sum_{i \in N} x_i^2 $$