Formatting Optimization Problems

Put the following lines in the head of your document, i.e. between \documentclass{xxx} and \begin{document}

\newcommand*{\optMinMax}[2]{\underset{\vphantom{A}\displaystyle\mathclap{#1}}%
                           {\text{#2}}\quad}
\newcommand{\optmin}[1]{\optMinMax{#1}{minimize}}
\newcommand{\optmax}[1]{\optMinMax{#1}{maximize}}
\newcommand*{\subject}{\text{subject to}\quad}
\newcommand*{\tagthis}{\stepcounter{equation}\tag{\theequation}}

\begin{align*}\label{eq:QP}
  \optmin{x}  &\dfrac{1}{2}x^THx + f^Tx \tagthis \\
  \subject    & Ax  \leq b \\
              &A_{\text{eq}}x  = b_{\text{eq}}
\end{align*}

\begin{align*}\label{Eq:Ign_Opt}
\optmin{\begin{matrix}\Delta\bm{\theta}_{\mathrm{SOI}},\Delta \theta_{\mathrm{HP}},\\
               \Delta \theta_{\mathrm{LP}},\Delta\theta_{\mathrm{Hot}}\end{matrix}}
         &\sum_{k=1}^{H_p}\biggl(\omega_1||r_{\theta_{50}}(k)-\bm{\theta}_{50}(k)||^2_2 
           + \omega_2 ||r_{\tau}(k)-\bm{\tau}(k)||^2_2  \\[-7mm]
          & \hphantom{\sum} \quad + \omega_3 \theta_{\mathrm{HP}}(k)^2 
            + \omega_4  \theta_{\mathrm{LP}}(k)^2\biggl)  + \cdots \tagthis\\
\subject  & l_b \leq \begin{pmatrix} \bm{\theta}_{\mathrm{SOI}} \\
                \theta_{\mathrm{Hot}} \\ \theta_{\mathrm{HP}} \\ \theta_{\mathrm{LP}} 
                \end{pmatrix} \leq u_b  \\
          & \text{other conditions} 
\end{align*}

\begin{alignat*}{2}\label{Eq:opt1}
  \optmin{\mathbf{u}_1 , \dots , \mathbf{u}_{H_p}} 
           & \sum_{k=1}^{H_p} J_{m_f}(k) + J_{p_{\text{IMEP}}}(k) + J_{\Delta u}(k) \hspace*{-40mm} &&\tagthis\\
  \subject & p_k \leq c_{p_{\text{max} }}              &&\forall\theta,\;k = 1, \ldots, H_p  \\
           & dp/d\theta_k \leq c_{dp_{\text{max}}}    &\quad& \forall\theta,\; k = 1, \ldots, H_p \\
           & \text{NO}_{x}(k) \leq c_{\text{NO}_x}     && k = 1, \ldots, H_p \\
           & T_{\text{ex}}(k) \geq c_{T_{\text{ex} }}  &&k = 1, \ldots, H_p^{T_{\text{ex}}} \\
           & \mathbf{u}(k) \in \mathbb{U}              &&k = 1, \ldots, H_p. 
\end{alignat*}