# Does every gutter point in SVM have positive multiplier?

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?

• Note that your SVM formulation is for data that can be separated, and that $b$ is an optimization variable. – LinAlg Nov 5 '16 at 22:31

• Appreciate the theoretical part, but in the context of SVM, do you know if it's possible that a gutter point has a $0$ multiplier? Just to make sure. – Nicholas Nov 5 '16 at 20:08