Complexity of an optimization problem with larger feasible set of an NP-hard problem

Question

Suppose the following problem is NP-hard:

\begin{eqnarray} \min & f(\boldsymbol{x})\\ s.t. & \\ A\boldsymbol{x} &= b\\ \boldsymbol{x} & \in Z^n \end{eqnarray}

Then what can we say about the complexity of the following problem:

\begin{eqnarray} \min & f(\boldsymbol{x})\\ s.t. & \\ B\boldsymbol{x} &= d\\ \boldsymbol{x} & \in Z^n \end{eqnarray} where $ \{ \boldsymbol{x} : A\boldsymbol{x} = b\} \subseteq \{ \boldsymbol{x} : B\boldsymbol{x} = d\}$ ?

Answer 1

In general, not much. Some extreme cases are

1. $B = A$ and $d=b$, in which case the complexities are the same too, and
2. $B=0$ and $d=0$, in which case the second problem is trivial.

The problems (more precisely, their decision versions) are in $NP$, because given $x$ you can verify the value of $f(x)$ and the constraint $Bx=d$ in polynomial time.