I was reading a class note on SVM from Andrew Ng (pp 19~20 from <http://cs229.stanford.edu/notes/cs229-notes3.pdf>) and can't understand something in the lecture note. It says that the L1-regularzed soft-margin problem is the following:

$$\begin{array}{ll}
\mbox{min.} & & \frac{1}{2} ||w||^2 + C \sum \xi\_i \\
 \mbox{s.t.} & & y\_i (w^T x\_i+b) \geq 1 - \xi\_i, i = 1,...,m\\
 & & \xi\_i \geq 0, i=1,...,m
\end{array}$$

It then says that the Lagrangian is 

$L(w,b,\xi, \alpha, r) = \frac{1}{2}w^Tw + C\sum{\xi\_i} - \sum \alpha\_i [y\_i (x^Tw+b)-1+\xi\_i] - \sum r\_i \xi\_i$

Then, it says that the KKT dual-complementarity condition are:
$$\begin{array}{ll}
\alpha\_i = 0 & \Rightarrow & y\_i (w^Tx\_i + b) \geq 1 \\
\alpha\_i = C & \Rightarrow & y\_i (w^Tx\_i + b) \leq 1 \\
0 < \alpha\_i < C & \Rightarrow & y\_i (w^Tx\_i + b) = 1 
\end{array}$$

I don't understand the last part. Isn't the KKT dual-complementarity condition $\alpha\_i [y\_i (x^Tw+b)-1+\xi\_i]$ = 0 -- that is, either $\alpha\_i = 0$ or $ y\_i (x^Tw+b)-1+\xi\_i = 0$. Can anybody explain to me how the above complementarity conditions are derived? I am troubled that $\xi\_i$ doesn't appear in the complementary condition.