To restate our problem: Given a linear separable dataset $\{\bar{x}_i, y^{(i)}\}_i^n$, and linear decision boundary defined by normal, offset $\bar{\theta}, b$ we would like to solve the following optimization problem with constraints:

$\max_{\bar{\theta}, b} \frac{\bar{\theta} \cdot \bar{x}_i + b}{||\bar{\theta}||}$ $\text{subject to } (y^{(i)} (\bar{\theta} \cdot \bar{x}_i + b)) > 0 \ \ \forall i$

This problem simplifies to $\min_{\bar{\theta}, b} \frac{1}{2}||\bar{\theta}||^2$ $\text{subject to } (y^{(i)} (\bar{\theta} \cdot \bar{x}_i + b)) \geq 1 \ \ \forall i$

The nitty-gritty of this proof can be seen in this nodes source, though it might be worth writing out in full here. (#TODO Connect the linear algebra relation node to this node for explanation).