Align objective function in latex

I think it's really a single line equation so I'd use equation not align, and then use aligned to wrap the subterm.

\documentclass{article}
\usepackage[cmex10]{amsmath}

\DeclareMathOperator*{\Min}{min}

\begin{document}

\begin{equation}
Q_t(v_{t-1},a_{ti\omega}) = 
\min_{
\substack{g_t,~y_t,f_t,~\theta_t,~\\\Delta u^{up},~\Delta u^{dn},\\
\Delta g^{up},~\Delta g^{dn},\\g_t^c,~y_t^c,~f_t^c,~\theta_t^c}}
c^\top_{t}g_{t,i}+{}
\begin{aligned}[t]
& \sum_{i \in \mathcal{I}^T}(c^{U}_i \Delta g^{up}_{t,i}+c^{D}_i \Delta g^{dn}_{t,i})+\\
&\sum_{i \in \mathcal{I}^H}(c^{U}_i \Delta u^{up}_{t,i}+c^{D}_i \Delta u^{dn}_{t,i})+\mathcal{Q}_{t+1}(v_t)
\end{aligned}
\end{equation}

\end{document}

And one more solution :-), based on use of multlined environment from package mathtools:

    \documentclass{article}
\usepackage[cmex10]{mathtools}

\usepackage[active,tightpage]{preview}
\PreviewEnvironment{align}
    \setlength\PreviewBorder{1em}

    \begin{document}
\begin{align}
Q_t(v_{t-1},a_{ti\omega}) =     \notag  \\
   \min_{\substack{g_t,~y_t,f_t,~\theta_t,~\\
          \Delta u^{up},~\Delta u^{dn},\\
          \Delta g^{up},~\Delta g^{dn},\\
          g_t^c,~y_t^c,~f_t^c,~\theta_t^c}}
   & \begin{multlined}[t][75mm]
   c^\top_{t}g_{t,i} +         
    \sum_{i \in \mathcal{I}^T}(c^{U}_i 
            \Delta g^{up}_{t,i}+c^{D}_i \Delta g^{dn}_{t,i}) + \\
    \sum_{i \in \mathcal{I}^H}(c^{U}_i 
            \Delta u^{up}_{t,i}+c^{D}_i \Delta u^{dn}_{t,i})+\mathcal{Q}_{t+1}(v_t)
        \end{multlined}
\end{align}
    \end{document}

Edit: I just discover stupid error in my code .. the position of \notag is on wrong side of \\. Now this is corrected and numbering of equation is consequently improved.

I'd use the multline environment instead of align. To reduce the spacing due to the long \substack, use \smash[b].

MWE

\documentclass{article}
\usepackage[cmex10]{amsmath}

\DeclareMathOperator*{\Min}{min}

\begin{document}

\begin{multline}
Q_t(v_{t-1},a_{ti\omega}) = 
\min_{\smash[b]{\substack{g_t,~y_t,f_t,~\theta_t,~\\\Delta u^{up},~\Delta u^{dn},\\\Delta g^{up},~\Delta g^{dn},\\g_t^c,~y_t^c,~f_t^c,~\theta_t^c}}} c^\top_{t}g_{t,i}+\sum_{i \in \mathcal{I}^T}(c^{U}_i \Delta g^{up}_{t,i}+c^{D}_i \Delta g^{dn}_{t,i})+ \\
\hspace{19mm}\sum_{i \in \mathcal{I}^H}(c^{U}_i \Delta u^{up}_{t,i}+c^{D}_i \Delta u^{dn}_{t,i})+\mathcal{Q}_{t+1}(v_t)
\end{multline}
\end{document}

Output