Trying to write optimization formulas, but I can't seem to align part of the restrictions

Question

In the following optimization problem:

\begin{align}
\text{Minimize} \quad
& \sum_{j=1}^{n} c_j x_j \\
\text{s.t.} \quad
& \sum_{j=1}^{n} f_{ij} x_j \geq 1 \quad i = 1 \ldots m \\
& \sum_{j=1}^{n} g_{kj} x_j \leq h_k \quad k = 1 \ldots K \\
& x_j \in \left \{ 0,1 \right \}
\end{align}

I would like to align

i = 1 \ldots m
k = 1 \ldots K

so they are similar to this:

I'm using \quad as a suggestion from ChatGPT. Any advice here will be extremely welcome and valuable.

Answer 1

You can specify the alignment via &.

Personally, I would not use any \quad but you may add them. In addition, I added commas. But these are just my recommendations. You might also want to have a look on https://ctan.org/pkg/lshort.

See also https://tex.stackexchange.com/a/31956/128553 for a different numbering scheme within the cases environment.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align}
\text{Minimize } 
    & \sum_{j=1}^{n} c_j x_j \
\text{s.t. } 
    & \sum_{j=1}^{n} f_{ij} x_j \geq 1,         & i = 1, \ldots, m, \
    & \sum_{j=1}^{n} g_{kj} x_j \leq h_k,       & k = 1, \ldots, K, \
    & x_j \in \left { 0, 1 \right }.
\end{align}
\begin{align}
\text{Minimize } 
    & \sum_{j=1}^{n} c_j x_j \
\text{s.t. } 
    & \begin{cases}
        \displaystyle \sum_{j=1}^{n} f_{ij} x_j \geq 1,         & i = 1, \ldots, m, \
        \displaystyle \sum_{j=1}^{n} g_{kj} x_j \leq h_k,       & k = 1, \ldots, K, \
        \displaystyle x_j \in \left { 0, 1 \right }.
    \end{cases}
\end{align}
\begin{align}
\text{Minimize } 
    & \sum_{j=1}^{n} c_j x_j \
\text{s.t. } 
    & \begin{cases}
        \sum_{j=1}^{n} f_{ij} x_j \geq 1,       & i &= 1, \ldots, m, \
        \sum_{j=1}^{n} g_{kj} x_j \leq h_k,         & k &= 1, \ldots, K, \
        x_j \in \left { 0, 1 \right }.
    \end{cases}
\end{align}
\end{document}