Background
Say you have a traditional linear-optimization problem, there is a linear cost function,
$\vec{c}\cdot\vec{x}$
and a set of linear constraints,
$A_1\vec{x} \geq b_1 $
$A_2\vec{x} \leq b_2 $
$A_3 \vec{x} = b_3$.
Inherently, the search space of the optimization problem is defined by a boolean AND of all the constraints.
Question
If the search space is discontinuous and the definition contains a boolean OR, is it still possible to reduce the problem to the standard form? If so, what is the method called? How does it work? Any references would be helpful.
I understand that the trivial solution would be to search each continuous subspace of the search space separately and do an iterative search through the discrete set of minima of all the subspaces. I was wondering if there is a more elegant solution.
Example
The following is an example of a constraint I am working with.
$(x = 0) \vee (x > x_{min})$ where $x_{min} > 0$
Note, that there may be several variables with a similar constraint, that the variable must be zero or larger than a minimum value.
As a linear programming problem, no. The feasible space of a linear program is an intersection of half spaces, and as such, convex. So if there is a feasible solution with $x = 0$ and a feasible solution with $x = y$ then there is a feasible solution with $x = \alpha y$ for every $0\le alpha\le 1$.
As a mixed integer program (a linear program where some variables are constrained to be integers) then, yes, you can constrain variables to discontinuous ranges. See the aimms guide for details. Note that while linear programming is easy (in P) integer programming is not (unless P=NP).