Alignment in the cases environment

Question

I have the following equation:

produced by:

\begin{equation}
    \text{Type I:}
    \begin{cases}
        \text{find      } &\vect{x} \\
        \text{minimizing} &\mu\left\{f_{i}(\vect{x}, \vect{p})\right\} \text{ and } \sigma\left\{f_{i}(\vect{x}, \vect{p})\right\} (i = 1, ...\, , n_{\text{objectives}}) \\
        \text{subject to} &L_{j}(\vect{x}, \vect{p}) \leq 0                                                                        (j = 1, ...\, , n_{\text{constraints}}) \\
                          &\vect{x}^{-} \leq \vect{x} \leq \vect{x}^{+}
    \end{cases}
\end{equation}

However, I want to align (i = ...) and (j = ...) in the second and third row, respectively, and add some spaces in front of them so that they are not that close to the mathematical equations, as illustrated in the image below. How to do that?

Answer 1

I suspect that you are after

but I'm not sure, since your code fragment doesn't produce showed images in your question.

Anyway, you can insert correct math expressions in the following skeleton:

\documentclass{article}
\usepackage{bm, mathtools}%

\begin{document}
\begin{align}
\text{find $\bm{x}$ which minimize}        
    &&&  \mu\left\{f_{i}(\bm{x}, \bm{p})\right\}     
        &&   (i = 1, \dotsc, n)     \nonumber    \\
\text{subject to} 
    &&& L_{j}(\bm{x}, \bm{p}) \leq 0 
        &&   (j = 1, \dotsc, n)                 \\
    &&& \bm{x}_a^{-} \leq \bm{x}_a \leq \bm{x}_a^{+}
        &&   (a = 1,2 \dotsc, n)    \nonumber 
\end{align}
\end{document}

For minimizing problems exist dedicated package optidef, but )m not familiar with it. For it use see for example @egreg answer here.

Answer 2

Here is a solution using the IEEEeqnarraybox from the IEEEtrantools package.

\documentclass{article}

\usepackage{geometry}
\usepackage{mathtools}
\usepackage{IEEEtrantools}

\begin{document}

\begin{IEEEeqnarray}{r;l}
    \text{Type I:} &
        \left\{
        \begin{IEEEeqnarraybox}[\relax][c]{;l'l'l}
            \text{find}       &\textbf{x}  \\
            \text{minimizing} &\mu\left\{f_{i}(\textbf{x}, \textbf{p})\right\} \text{ and } \sigma\left\{f_{i}(\textbf{x}, \textbf{p})\right\} &(i = 1, ...\, , n_{\text{objectives}}) \\
            \text{subject to} &L_{j}(\textbf{x}, \textbf{p}) \leq 0                                                                            &(j = 1, ...\, , n_{\text{constraints}}) \\
                              &\textbf{x}^{-} \leq \textbf{x} \leq \textbf{x}^{+}
        \end{IEEEeqnarraybox}
        \right.\IEEEeqnarraynumspace
\end{IEEEeqnarray}
\end{document}