# Support vector machine, complementary slackness and marginal hyperplane

One of the complementary slackness conditions for a support vector machine states that $$\alpha_i ( y_i (\langle w, x_i \rangle + b ) -1 ) = 0,$$ where $$\alpha_i$$ is the lagrange variable. One can then conclude that if $$\alpha_i \neq 0$$ ($$x_i$$ is a support vector), $$y_i (\langle w, x_i \rangle + b ) -1$$ ($$x_i$$ lies on the marginal hyperplane). My question is if one can say anything in the other direction; if a point $$x_i$$ lies on the marginal hyperplane, can one conclude that it is a support vector? ($$\alpha_i\neq 0$$).

Theoretically, no. You can see this in terms of smoothly adjusting $$C$$ - at some point $$x_i$$ may be on the margin but with $$\alpha_i = 0$$. Practically, yes. In particular, note that stopping conditions for the optimization allow for some $$\epsilon$$, generally in terms of the objective, so you could certainly have a "small enough" value for $$\alpha_i$$ for these points. Practically speaking though, when you actually solve the optimization problem, you will see that not all non-bound SVs ($$0 < \alpha_i < C$$) actually aren't classified as $$\pm 1$$, but rather are close to this value. This is why e.g., libSVM calculates the bias term, $$b$$, by averaging over the classification values for these points (see e.g. 4.1.5 in the libSVM paper)