When do equality and inequality constrained problems agree?

Question

Let $f:\mathbb{R}^d\to\mathbb{R}$ and $h:\mathbb{R}^d\to\mathbb{R}$. Consider a solution $x^\ast$ to the equality constrained optimisation problem $$ x^\ast \in \underset{x}{\text{argmin}} f(x) \quad\text{subject to} \quad h(x) = 0,$$ supposing at least one exists. What are sufficient conditions that guarantee that $x^\ast$ is also a solution to the corresponding inequality constrained problem, so that $$ f(x^\ast) = \underset{y}{\text{min}} f(y) \quad\text{subject to} \quad h(y) \leq 0?$$