How to label each equation in align environment?

You can label each line separately, in your case:

\begin{align}
  \lambda_i + \mu_i = 0 \label{eq:1}\\
  \mu_i \xi_i = 0 \label{eq:2}\\
  \lambda_i [y_i( w^T x_i + b) - 1 + \xi_i] = 0 \label{eq:3}
\end{align} 

Note that this only works for AMS environments that are designed for multiple equations (as opposed to multiline single equations).