Starting from $$\hat{\beta} = \arg \min_\beta \|X\beta - y\|_2^2 \text{ s.t. } (1-\alpha)\|\beta\|_1 + \alpha\|\beta\|_2^2 \leq t,$$ we can write the dual Lagragian formulation of this optimization problem as $$ \begin{array}{rcl} L(\beta,\alpha,\lambda) & = & \|X\beta - y\|_2^2 + \lambda \left( (1-\alpha)\|\beta\|_1 + \alpha\|\beta\|_2^2 - t\right) \\ & = & \|X\beta - y\|_2^2 + \lambda (1-\alpha)\|\beta\|_1 + \lambda\alpha\|\beta\|_2^2 - \lambda t, \end{array} $$ and we see that this indeed looks like the first problem that you wrote, with parameters $\lambda_1$$\lambda_1=\lambda (1-\alpha)$ and $\lambda_2$ with an additional$\lambda_2=\lambda \alpha$, which leads to the expression of the "elastic" link between these two parametersparameter: $$\alpha = \frac{\lambda_2}{\lambda_1+\lambda_2}.$$ That being said, to go from this point to Zou and Hastie's assertion that both problems are equivalent, I admit that I miss a step or two...
Starting from $$\hat{\beta} = \arg \min_\beta \|X\beta - y\|_2^2 \text{ s.t. } (1-\alpha)\|\beta\|_1 + \alpha\|\beta\|_2^2 \leq t,$$ we can write the dual Lagragian formulation of this optimization problem as $$ \begin{array}{rcl} L(\beta,\alpha,\lambda) & = & \|X\beta - y\|_2^2 + \lambda \left( (1-\alpha)\|\beta\|_1 + \alpha\|\beta\|_2^2 - t\right) \\ & = & \|X\beta - y\|_2^2 + \lambda (1-\alpha)\|\beta\|_1 + \lambda\alpha\|\beta\|_2^2 - \lambda t, \end{array} $$ and we see that this indeed looks like the first problem that you wrote, with parameters $\lambda_1$ and $\lambda_2$ with an additional "elastic" link between these two parameters: $$\alpha = \frac{\lambda_2}{\lambda_1+\lambda_2}.$$ That being said, to go from this point to Zou and Hastie's assertion that both problems are equivalent, I admit that I miss a step or two...
Starting from $$\hat{\beta} = \arg \min_\beta \|X\beta - y\|_2^2 \text{ s.t. } (1-\alpha)\|\beta\|_1 + \alpha\|\beta\|_2^2 \leq t,$$ we can write the dual Lagragian formulation of this optimization problem as $$ \begin{array}{rcl} L(\beta,\alpha,\lambda) & = & \|X\beta - y\|_2^2 + \lambda \left( (1-\alpha)\|\beta\|_1 + \alpha\|\beta\|_2^2 - t\right) \\ & = & \|X\beta - y\|_2^2 + \lambda (1-\alpha)\|\beta\|_1 + \lambda\alpha\|\beta\|_2^2 - \lambda t, \end{array} $$ and we see that this indeed looks like the first problem that you wrote, with parameters $\lambda_1=\lambda (1-\alpha)$ and $\lambda_2=\lambda \alpha$, which leads to the expression of the "elastic" parameter: $$\alpha = \frac{\lambda_2}{\lambda_1+\lambda_2}.$$ That being said, to go from this point to Zou and Hastie's assertion that both problems are equivalent, I admit that I miss a step or two...
Starting from $$\hat{\beta} = \arg \min_\beta \|X\beta - y\|_2^2 \text{ s.t. } (1-\alpha)\|\beta\|_1 + \alpha\|\beta\|_2^2 \leq t,$$ we can write the dual Lagragian formulation of this optimization problem as $$ \begin{array}{rcl} L(\beta,\alpha,\lambda) & = & \|X\beta - y\|_2^2 + \lambda \left( (1-\alpha)\|\beta\|_1 + \alpha\|\beta\|_2^2 - t\right) \\ & = & \|X\beta - y\|_2^2 + \lambda (1-\alpha)\|\beta\|_1 + \lambda\alpha\|\beta\|_2^2 - \lambda t, \end{array} $$ and we see that this indeed looks like the first problem that you wrote, with parameters $\lambda_1$ and $\lambda_2$ with an additional "elastic" link between these two parameters: $$\alpha = \frac{\lambda_2}{\lambda_1+\lambda_2}.$$ That being said, to go from this point to Zou and Hastie's assertion that both problems are equivalent, I admit that I miss a step or two...