# What is the easiest method to solve a regularized SVM with Lagrangian?

Consider the regularized SVM approach: $$$$g (\alpha) = \sum_{i=1}^{n} \alpha_i - \frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n} \alpha_i \alpha_j y_i y_j \langle \phi(x_i), \phi(x_j)\rangle_H$$$$ under the constraints $$0\leq \alpha_i \leq C$$ for $$i=1, \dots, n$$ and $$\sum_{i=1}^{n} \alpha_i y_i = 0$$, where $$\phi$$ is some feature-map into a Hilbert space $$H$$.

What is the easiest method to solve this? Is something like the SOM-Algorithm necessarily needed?

If one tries to solve this analytically (setting the gradient to zero and solving the system of n linear equations), it doesn't work because the parameter $$C$$ does not appear...