My question is about the following paper:
http://webdocs.cs.ualberta.ca/~maz/publications/ratliff_nathan_2007_3.pdf
In section 2 they show Equation 3 (which is just an optimization problem), which they transform into the optimization problem right before section 3. Everything is very easy to read and self-contained. However, I do not understand how they can tell so easily that the constraints in the convex program are tight (which allows them to transform the program to the one before section 3). Are they relying on some result that they are not saying, or is this a very simple argument coming from optimization?
An edit, following a request to rewrite the question in a self-contained way. Let $w \in R^d$ and $y_i \in R^d$ and $A_i$ be a finite set of vectors in $R^d$. Consider the following optimization problem:
$\min_{w,\xi} ||w||_2^2 + \sum_i \xi_i$
under the constraints that $\forall i$ we have $w^T y_i + \xi_i \ge \max_{y \in A_i} \left( w^T y \right)$
why in any solution of this optimization problem, the constraints are tight, so that we can reformulate the optimization problem by replacing $\xi_i$ with $\max_{y \in A_i} \left(w^T y \right) - w^T y_i$ and get rid of the constraints?
