I have a function to be maximized subject to constraints. I can write the primal Lagrange function as the following: (objective function WITH two constraints in the last two terms)
$$L_P = \frac{1}{2} ||\beta||^2 + C\sum_{i=1}^N \xi_i - \sum_{i=1}^N\alpha_i[y_i(x_i^T\beta + \beta_0)-(1-\xi_i)] - \sum_{i=1}^N \mu_i \xi_i$$
So, as usual, I took the partial derivatives $\frac{\partial L_P}{\partial \beta} = 0$, $\frac{\partial L_P}{\partial \beta_0} = 0$ and $\frac{\partial L_P}{\partial \xi_i} = 0$, and we have the following results:
$$\beta = \sum_{i=1}^N \alpha_iy_ix_i$$
$$0 = \sum_{i=1}^N \alpha_iy_i$$
$$\alpha_i = C - \mu_i $$
Finally, I attempted to get the last three equations and plug them into $L_P$ to complete the optimization, and I don't know what to do anymore. The final result should be:
$$L_D = \sum_{i=1}^N \alpha_i - \frac{1}{2}\sum_{i=1}^N \sum_{i'=1}^N\alpha_i \alpha_{i'}y_iy_{i'}x_i^Tx_{i'}$$