Given a primal problem
$$ \min_{x \in \mathbf{R}^2} \{ f(x)|g(x)=0 \}
$$
that has a solution $f(x^*)=0$ (where $x^*$ is the value of x that minimizes $f$ such that $g(x)=0$),
and given a dual function
$$ \mathcal{D}(\lambda) = \inf_{x \in \mathbb{R}^2} \mathcal{L}(x, \lambda) = -\infty$$
where $\mathcal{L}$ is the Lagrangian, we have that the Dual Problem is:
$$
\sup \mathcal{D}(\lambda) = -\infty \quad \text{(I think)}
$$
Does this imply that the duality gap is $\infty$?
Note: the infimum of the Lagrangian can be $-\infty$ if $f$ is non-convex.