How to prove the following two optimization problems are equivalent?

Question

The first problem is $$\min_{\mathbf{x}\subset\mathbb{R}^n\backslash\{0\}}\|\mathbf{x}\|_1-0.3\|\mathbf{x}\|_p\quad\text{s.t.}\quad\|\mathbf{A}\mathbf{x}-\mathbf{y}\|_2\leq\varepsilon$$ where $\mathbf{A}$ and $\mathbf{y}$ are fixed matrix/vector.

The second problem is $$\min_{\mathbf{x}\subset\mathbb{R}^n\backslash\{0\}}\frac{1}{2}\|\mathbf{A}\mathbf{x}-\mathbf{y}\|_2^2+\lambda(\|\mathbf{x}\|_1-0.3\|\mathbf{x}\|_p)$$

Answer 1

Hint 1:

What is the Lagrangian function of the first problem?

Hint 2:

What are the KKT conditions of that Lagrangian?

Hint 3:

How do the optimal solutions to both problems correlate?