How are Lagrangian multipliers zero except for support vectors in dual representation of SVM?

Question

How can we conclude that the Lagrangian multipliers are zero, except support vectors, in a dual problem? I cannot seem to see it.

$$L(\alpha)=-\frac{1}{2}\sum_i \sum_j \alpha_i \alpha_j y_i y_j x_i' x_j + \sum_i{\alpha_i} $$

Answer 1

In optimization, we have something called complementary slackness condition, it is part of the KKT conditions.

Every constraint, $g_i(x^*)\le 0$ in the primal corresponds to a dual variable $\mu_i$ (Lagrange multiplier). The condition state that $$g_i(x^*)\mu_i=0$$

For points that are not support vectors, we have $g_i(x^*)<0$, hence we must have $\mu_i=0$.