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.