I understand that SVM is about solving the constrained optimization such that
$$\min_{\mathbf{w}} \dfrac{1}{2} \mathbf{w}^T\mathbf{w}$$ subject to $$y_i(\mathbf{w}^T\mathbf{x_i}+b)\geq{1}, i=1, 2, ...,n$$
And this is handled using nonlinear optimization method Karush–Kuhn–Tucker approach where one step is based on the necessary complementary slackness condition such that
$${\alpha}_i\left(y_i(\mathbf{w}^T\mathbf{x_i}+b)-1\right)=0, i=1, 2, ...,n$$ has to be satified.
Because for non-gutter dots (i.e., the points not on the edge of the separating hyperplane), we have
$$y_i(\mathbf{w}^T\mathbf{x_i}+b)-1 > 0$$
the corresponding multiplier $\alpha_i$ then must be $0$. But my question is for gutter points, because
$$y_i(\mathbf{w}^T\mathbf{x_i}+b)-1 = 0$$
we know that the corresponding multiplier $\alpha_i$ should be non-negative, but are they necessarily positive? In other words, if I define support vector as any $\mathbf{x_i}$ on the gutter, then is this the same as if I define support vector as any $\mathbf{x_i}$ whose multiplier is positive?