I am wondering if the set of optimizers of the problem
$$ \min_{x \in X} \ f(x) \quad \text{subject to: } g(x) \leq 0, \ h(x) = 1 $$
is the same of the one of
$$ \min_{x \in X} \ f(x) + h(x) \quad \text{subject to: } g(x) \leq 0, \ h(x) = 1, $$
given that the quantity $h(x)$ is anyways constant.
Minimizing $f$ should be the same of minimizing $f+1 = f+h$. Not sure am I right.