Formatting Optimization Problems

Put the following lines in the head of your document, i.e. between \documentclass{xxx} and \begin{document}. The package mathtools is an extension of amsmath, and may be of interest independently of optimization problems.

\usepackage{mathtools}
\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}}

This is a simple optimization. Note that the equation number is put at the expression to be minimized, using the command \tagthis. For other numbering patterns, see below. Note also that the first column, with 'minimize' and 'subject to', is right aligned, and the second column is left aligned. Other alignments are possible, but this is the simplest and most estetic in my opinion.

\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*}

This is a more complex optimization. Note that the expressions under 'minimize' is in fact wider than the word 'minimize'. This is acheived through the command \clap, which typesets its arguments centered and with an apparent width of 0pt. It is therefore up to yourself to allow a reasonable extra width. This command is available through the package 'mathtools'. Note also that the \tagthis command is on the second line, because that line is a continuation line.

\begin{align*}\label{Eq:Ign_Opt}
\optmin{\begin{matrix}\Delta\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)-\theta_{50}(k)||^2_2 
             + \omega_2 ||r_{\tau}(k)-\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} \theta_{\mathrm{SOI}} \\
               \theta_{\mathrm{Hot}} \\ \theta_{\mathrm{HP}} \\ \theta_{\mathrm{LP}} 
               \end{pmatrix} \leq u_b  \\
           & \text{other conditions} 
\end{align*}

This is another fairly complex optimization, that requires two columns for the description of the conditions.

\begin{alignat*}{2}\label{Eq:opt1}
\optmin{\mathbf{u}_1 , \dots , \mathbf{u}_{H_p}} 
         & \mathrlap{\sum_{k=1}^{H_p} J_{m_f}(k) + J_{p_{\text{IMEP}}}(k) + J_{\Delta u}(k)}  &&\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*}