Tight Against the Boundary

In the solution to a hard-margin SVM's dual problem, a training point $x_i$ has a Lagrange multiplier $\alpha_i > 0$ . According to the KKT complementary slackness condition, what must be true about $x_i$ ?

$x_i$ must lie strictly inside the margin on the correct side, contributing a nonzero slack penalty

$x_i$ can be any training point, since $\alpha_i$ is unconstrained by the position of $x_i$ relative to the margin

$x_i$ must lie exactly on one of the two margin boundaries, meaning it is a support vector satisfying $y_i(w \cdot x_i + b) = 1$

$x_i$ must be misclassified by the final decision boundary

$x_i$ must be the single point closest to the class centroid, unrelated to the margin boundary