# Non-convex linear program optimisation with infinite number of OR constraints

I am aware that when we have a linear problem subject to OR constraints, the LP would be a non-convex optimisation problem. For example,

$${x = 0}$$ OR $${1<=x<=2}$$.

My question is in such a situation, is it possible to check if an LP is feasible/infeasible, bounded/unbounded or if it has an optimal value or not

I could not find much explanation on the internet concerning a detailed explanation of this situation. I'd appreciate it if anyone could explain this in more detail.

Similar questions in other sites:

https://math.stackexchange.com/questions/4158912/infinite-number-of-or-constraints-in-linear-programming

https://stackoverflow.com/questions/50987517/expressing-an-or-constraint-in-linear-programming

• What exactly is your question? I don't see a question here. We are a question-and-answer site, we require you to articulate a specific question that you want answered. When you talk about "a detailed explanation of this situation", I'm not sure what you are hoping for.
– D.W.
Jun 10, 2021 at 5:03
• Hi @D.W. I have edited the problem description. Apologies for the vagueness Jun 10, 2021 at 10:49
• math.stackexchange.com/q/4163035/14578
– D.W.
Jun 12, 2021 at 4:31

Yes, it's possible. Let $$S$$ be the original linear problem, i.e., a conjunction of inequalities. You can form two LPs. One LP has the form $$S \land x=0$$, which can be solved with a LP solver. The other LP has the form $$S \land 1 \le x \le 2$$. If either one has a feasible solution, your problem has a feasible solution. This procedure takes polynomial time.