Aligning split equations to the left perfectly

This solution uses array. Like tabular, it adds a fixed space between columns. Just for fun, I also lined up the inequalities.

\documentclass{article}
\usepackage{mathtools}%for \vdotswithin
\usepackage{breqn}% I have no idea what this does

\begin{document}

\begin{equation}
\begin{array}{ll@{}l}
\text{minimize} &Z(w)\\
\text{subject to} &f_1(w) &{}\leq a_1\\
& f_2(w) &{}\leq a_2 \\
&&\vdotswithin{\leq} \\
&f_m(w) &{}\leq a_m,\\
\end{array}
\end{equation}

\end{document}

Output:

Here is a possible solution:

% My standard header for TeX.SX answers:
\documentclass[a4paper]{article} % To avoid confusion, let us explicitly 
                                 % declare the paper format.

\usepackage[T1]{fontenc}         % Not always necessary, but recommended.
% End of standard header.  What follows pertains to the problem at hand.

\usepackage{mathtools} % also loads amsmath



\begin{document}

Text before the equation
\begin{equation}
    \begin{alignedat}{2}
        & \text{Minimize } && Z(w) \\
        & \text{subject to } && f_{1}(w)\le a_{1} \\
        & && f_{2}(w)\le a_{2} \\
        & && \hphantom{f_{2}(w)}\vdotswithin{\le} \\
        & && f_{m}(w)\le a_{m} \\
    \end{alignedat}
    \label{eq:minimize}
\end{equation}
Text after the equation.

\end{document}

Output:

Just make it simple, the & is your friend here, move the text to other side.

\documentclass{article}
\usepackage{amsmath}
\usepackage{breqn}

\begin{document}

\begin{equation}
\begin{split}
\text{minimize \quad}&Z(w) \\
\text{subject to\quad }&f_1(w) \leq a_1\\
&f_2(w) \leq a_2 \\
&\quad\quad \vdots\\
&f_m(w) \leq a_m,
\end{split}
\end{equation}


\end{document}