Consider the the optimization program of the kernelized SVM:
$$\text{maximize}_{\alpha} ~~ \alpha^T1-\alpha^TQ\alpha$$ $$\text{subject to:} \sum_{i=1}^N \alpha_iy_i=0,~0\leq \alpha_i\leq c$$
where for the matrix $Q$, we have: $Q_{ij}=y_iy_j\text{kernel}(x_i,x_j)$.
This optimization program is resulted by taking the dual of the hinge-loss-based SVM with $L_2$ regularization.
Now here is my question: Why no body regularizes $\alpha$ in this optimization program?
I understand that $\alpha$ is just the dual variables' vector, but I'm really curious to know why people have not tried regularizing $\alpha$.
For example, if for what ever reasons we want to force the algorithm to introduce less support vectors, then adding an $L_1$ regularization on $\alpha$ would be usefull right?
I.e.,
$$\text{maximize}_{\alpha} ~~ \alpha^T1-\alpha^TQ\alpha-\|\alpha\|_1$$ $$\text{subject to:} \sum_{i=1}^N \alpha_iy_i=0,~0\leq \alpha_i\leq c$$
Or maybe, a Tikhonov regularization:
$$\text{maximize}_{\alpha} ~~ \alpha^T1-\alpha^T(Q+\gamma I)\alpha$$ $$\text{subject to:} \sum_{i=1}^N \alpha_iy_i=0,~0\leq \alpha_i\leq c$$
I appreciate it if you share any comments that you might have.