According to Hastie's paper, the elastic net has two equivalent formulations:
$$\hat{\beta} = \underset{\beta}{\operatorname{argmin}} \left\{ \sum_{i=1}^N\left(y_i-\sum_{j=1}^p x_{ij} \beta_j\right)^2 + \lambda_1 \sum_{j=1}^p |\beta_j|+ \lambda_2 \sum_{j=1}^p \beta_j^2 \right\}$$
and
$$\hat{\beta} = \underset{\beta}{\operatorname{argmin}} \left\{ \sum_{i=1}^N\left(y_i - \sum_{j=1}^p x_{ij} \beta_j\right)^2\right\} \;\text{ s.t. } \;(1-\alpha)\sum_{j=1}^p |\beta_j| + \alpha\sum_{j=1}^p \beta_j^2 \leq t$$
where $\alpha = \frac{\lambda_2}{\lambda_1 + \lambda_2}$
My question is how to prove this equivalence formally. Ridge regression and the lasso also have these two possible formulations, but I could not find any reference where this equivalence is proven. A similar question I found in CrossValidated is this one
Lagrangian relaxation in the context of ridge regression
but I'm unable to understand Tristan's explanation. I have some understanding of Lagrange optimization theory, and I guess the answer is around those lines, but since all the papers treat the equivalence as obvious I would like to find a proper reference where this is explicitly demonstrated.