Understanding the constraints of SVMs in the non-separable case?

Question

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.

Answer 1

So I have found in another book that, introducing the slack variable

$\min_{w,b,\xi} \frac{1}{2}||w||^2+C\sum^N_{i=1}\xi_i$

s.t.

$t_iy(x_i)\geq1-\xi_i$ and $\xi_i\geq0$

is essentially a rewrite of the hinge+regularizer loss,

$\min_{w,b} \frac{1}{2}||w||^2+C\sum^N_{i=1}max(0, 1-t_iy(x_i))$.

So the slack variable $\xi$ is implicitly defined as $max(0, 1-t_iy(x_i))$ by the constraints.