Why is the constraint for SVM with offset the way it is? $\sum^{n}_{t=1}\alpha_ty^{(t)} = 0$

Question

When doing the dual formulation of the SVM we get the lagrangian:

$$L(\theta, \alpha) = \frac{1}{2}||\theta||^2 + \sum_{t=1}^n{ \alpha_t(1-y^{(t)}(\theta \cdot x^{(t)} + \theta_0) )}$$

and then by lagrange multipliers we get that one of the constraints is:

$$\alpha_t \geq 0$$

However, I am not sure if I understand where the additional constraint:

$$\sum^{n}_{t=1}\alpha_ty^{(t)} = 0$$

Comes from. How does one derive that additional constraint?

Answer 1

Take the derivative of $L(\theta,\alpha)$ with respect to $\theta_0$ and set it to zero will give the constraint you are asking about.