In Pattern Recognition and Machine Learning Section 7.1:
Based on what I understood so far, the slack variable $\xi$ is defined as $max(0, 1-t_ny(x_n))$ and it's associated with the hinge loss.
However it seems to me that the two constraints $t_ny(x_n)\geq1-\xi_n$ and $\xi_n\geq0$ are just two properties of $\xi$ according to on how it is defined, and without them it is still a valid optimization problem to solve (hinge loss + regularizer).
Why do we want to use them as the constraints again?
Or the slack variable is not explicitly defined as $max(0, 1-t_ny(x_n))$ but is only defined by the constraints?
Please correct me where I'm wrong.