Is local optima of QUBO can be feasible to its ILP problem, if the global optima of QUBO isn't feasible to its ILP?

Question

I'm new to optimization and I'm reading a paper that they reformulate an ILP problem into QUBO using penalty method.

Let's say we use a solver to solve above QUBO, and the global optima of QUBO obtained from the solver is infeasible to the original ILP. (infeasible from my understanding is the QUBO solution does not satisfy all constraints of the ILP).

My dumb question is: Can the local optima of QUBO obtained from the solver still be feasible to the original ILP?

I don't have much background in optimization, so it would be nice if you guys can suggest some materials mentioning my question. Thank you in advance!

Answer 1

You are correct that infeasible means one or more original constraints are violated. Assuming (without loss of generality) that you are minimizing in both the ILP and QUBO, the QUBO will have a convex objective function, meaning any local optimum is also global. So the answer to your question is no.

Assuming the original ILP is bounded, you can avoid getting an infeasible solution as the QUBO optimum by cranking up the penalties enough.