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Consider the regularized SVM approach: \begin{equation} 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 \end{equation} 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...

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