To relax the optimization problem for Hard-Margin SVM, we allow data points to live inside of the margin for the decision boundary, however we introduce a penalty term $\xi_i$. We then reformulate the problem as follows: $\min_{\bar{\theta}, b, \xi_i} \frac{1}{2}||\bar{\theta}||^2 + C\sum_{i=1}^N\xi_i$ $\text{subject to } y^i (\bar{\theta} \cdot \bar{x}^i + b) \geq 1 - \xi_i, \xi_i > 0 \forall i$

You might notice that this formula is noticeably different than what is shown in the book. However, this reformulation will lead us down a very interesting rabbit hole.