Under ridge regression, we seek to find estimates of the parameters $(\beta_0, \beta_1,\ldots,\beta_p)$ of an MLR model by either:
(1) Minimizing the objective function $$ f(\beta)=\sum_{i=1}^{n}(y_i-\hat{y_i})^2+\lambda\sum_{i=1}^p\beta^2_i $$
where $\hat{y_i}=\hat{\beta_0} + \hat{\beta_1}x_{i1}+\cdots+\hat{\beta}_px_{ip}$ and $\lambda \geq 0$.
(2) Minimizing
$$ \sum_{i=1}^{n}(y_i-\hat{y_i})^2 $$
subject to $\sum_{i=1}^p\beta^2_i \leq s$, where $s$ is some pre-specified positive number.
I was wondering if there was a nice way we can express $s$ in terms of $\lambda$, or vice versa.