Non-convex constraint made cost

Question

Consider the non-convex optimization problem

$$ \min_{x \in X} \ f(x) \quad \text{s.t.:} \ \ g(x) \leq 0, \ h(x) = 0 $$

where $X \subset \mathbb{R}^{2n}$ is compact and convex, $f$ and $g$ are convex, and

$$ h(x) = \left( \begin{array}{c} x_1^2 + x_2^2 - 1 \\ x_3^2 + x_4^2 - 1 \\ \vdots \end{array}\right). $$

Now consider the optimization problem

$$ \min_{x \in X} \ f(x) + \lambda \ \|h(x)\| \quad \text{s.t.:} \ \ g(x) \leq 0$$

for some $\lambda > 0$.

I am wondering if:

1. for $\lambda$ large enough, the two optimal solutions get arbitrarily close;

2. there exists $\lambda$ large enough such that the two optimal solutions coincide.