Ridge Regression objective$$\underset{\beta}{\text{min}} \sum_{i=1}^n (y_i - \beta \cdot x_i)^2 + \lambda \|\beta\|_2^2$$
SVM primal problem:
$$\begin{align} \max_{\mathbf{\alpha}} \quad &\min_{\mathbf{w},b} \frac{\|\mathbf{w}\|}{2}+ C \sum_{i=1}^{N} \alpha^{(i)} \left(1-\mathbf{w^T}\phi \left(\mathbf{x}^{(i)}\right)+b)\right), \\ s.t. \quad&0 \leq \alpha^{(i)} \leq C, &\forall i \in \{1,\dots,N\} \end{align}$$
Why does SVM have a hyperparameter parameter C on the hinge-loss function while in Ridge Regression There's No C parameter infront of the quadratic loss function?
Likewise, why is there no $\lambda$ parameter behind $||w||$ in SVM so $\lambda$ is assumed 1/2 in SVM?
Why isn't ridge: $$\underset{\beta}{\text{min}} \ C\sum_{i=1}^n (y_i - \beta \cdot x_i)^2 + \lambda \|\beta\|_2^2$$
Why isn't the SVM:
$$\begin{align} \max_{\mathbf{\alpha}} \quad &\min_{\mathbf{w},b} \lambda\|\mathbf{w}\|+ C \sum_{i=1}^{N} \alpha^{(i)} \left(1-\mathbf{w^T}\phi \left(\mathbf{x}^{(i)}\right)+b)\right), \\ s.t. \quad&0 \leq \alpha^{(i)} \leq C, &\forall i \in \{1,\dots,N\} \end{align}$$?