floor/ceiling/round functions in the constraints of an optimization?

Question

I have a constrained optimization problem in which I have to impose a "floor" or "ceiling" constraint to the solution. In fact I decided to use these nonlinear rounding functions because I needed to have an integer solution (or sometimes multiple of a specific scalar). My problem can be simplified as:

$D=||AX+B||_2$

minimize $D,$

s.t. $x_1=round(x_1), x_2=round(x_2), ... ,x_n=round(x_n)$

I know this notation is not perfect! I just wanted to mathematically show what I need.

So my question is that: Is there any way to have such constraints in an optimization? A "solvable!" problem of course, because I'm no expert in this field.

Thanks in advance!

Answer 1

You could look into integer programming which can be used to solve linear problems (yours is quadratic) where the variables are restricted to be integers. Please beware that this is not a convex problem and that it's NP-hard. GLPK can be used to solve these problems.