Recently, I have been dealing with an integer program that involves finding a $4$-level integral solution
$$x \in \{-2, -1, 1, 2\}$$
or even $x \in \{\pm n, \pm n-1, \dots, \pm 1\}$ which is multilevel integers solution. The cost function is quadratic and $x$ is the only optimization variable.
I search the network but I could not find useful information about this kind of problem. So is there any reference (algorithm) for solving this kind of problem? Could anyone provide some suggestions? Thanks in advance!
Add constraint:
$x \in \{1, -1, +1i, -1i\}$ or with even more levels with unitary amplitude $|x| = 1$.
One way to restrict $x$ to a discrete set $\lbrace a_1,\dots,a_n\rbrace$ is to create $n$ binary variables $z_1,\dots,z_n$ and add the constraints $$x=\sum_{i=1}^n a_i z_i$$ and $$\sum_{i=1}^n z_i =1.$$