Long equation in alignat with omit and rlap running off the edge

Question

I recently learnt the best(?) way to format an linear program in LaTeX is something like this:

\begin{alignat}{3}
  \text{min:} \quad & \omit{\rlap{$ \sum_{i=1}{n} c_i X_i $}} \\
  \text{s.t.} \quad & a_t \leq \sum_{i=1}{n} X_i \leq b_t \quad && t=1,\cdots,T \\
                    & X_i \in \{0,1\}                     \quad && i=1,\cdots,n
\end{alignat}

This way, the constraints are all left-aligned after the "s.t." with a little space, all the constraint domains are left-aliged after the longest constraint with a little space, and even if the objective function is long(ish) then I don't get a huge gap as I would if I just put a && at the end without the \omit{\rlap{$[...]$}}.

Anyway, the above is just some nonesense I wrote. But here is an actual one that is misbehaving in my document:

\begin{alignat}{3}
    \text{min:} \quad & \omit{\rlap{$\alpha\sum_{i=1}^N\sum_{k=1}^K\sum_{l=1}^K b_{k,l}Y_{i,k}Y_{i,l} + \beta\sum_{i=1}^N\sum_{j=1}^M c_{i,j}X_{i,j} + \gamma(\sum_{j=1}^M S_j + T_j)$}} \\
    \text{s.t.} \quad & \sum_{i=1}^N X_{i,j} + S_j - T_j = 1 \quad && j=1,\ldots,M \\
                      & S_j \in \mathbb{Z}^+                 \quad && j = 1, \ldots, M \\
                      & T_j \in \mathbb{Z}^+                 \quad && j = 1, \ldots, M
\end{alignat}

When I build this, the objective function (the first line) runs off the end of the page, however there is plenty of space on the left to fit it in.

I tried adding an extra empty alignment point & at the end of each line, and after the end of the \omit, but that did absolutely nothing.

I can fix it by splitting the first line into two, but I'd prefer to just move the whole thing to the left.

Thanks.

Edit 1: actually I didn't even notice this before, but it looks like the first equation is in-line math in text (which makes sense since it's in $). The parameters of the sums are not above and below like in the other lines, but next to them. Any idea how to fix this?

Edit 2: ok, \displaystyle after the $ fixes the problem with the first line being in-line math mode.

Answer 1

Add some space by hand, in the code below it's \hspace{9em}.

I made also some small changes: a macro for the hidden equation and \bigg for the parentheses around the summation. The \ldots should be \dots.

\documentclass{article}
\usepackage{amsmath,amssymb}
\usepackage{showframe} % just for the example

\newcommand{\eqhide}[1]{%
  \omit\makebox[0pt][l]{$\displaystyle#1$}%
}

\begin{document}

\begin{alignat}{2}
\text{min:} \quad & \eqhide{\alpha\sum_{i=1}^N\sum_{k=1}^K
                            \sum_{l=1}^K b_{k,l}Y_{i,k}Y_{i,l} +
                            \beta\sum_{i=1}^N\sum_{j=1}^M c_{i,j}X_{i,j} +
                            \gamma\biggl(\,\sum_{j=1}^M S_j + T_j\biggr)} \\
\text{s.t.} \quad & \sum_{i=1}^N X_{i,j} + S_j - T_j = 1 \quad && j = 1 ,\dots, M \hspace{9em} \\
                  & S_j \in \mathbb{Z}^+                 \quad && j = 1, \dots, M \\
                  & T_j \in \mathbb{Z}^+                 \quad && j = 1, \dots, M
\end{alignat}

\end{document}

Answer 2

Here is a solution without \rlap, nor \omit, which induces the bounds of sums are types at the right side of the sym symbol, but with a better use of the &. B.t.w. please note 3 alignment points require 5&. I also propose to use the subequations` environment, to have the main equation number for the equation itself (1st line) and 3 subsidiary counters for the constraints:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{showframe}%
 \renewcommand{\ShowFrameLinethickness}{0.3pt}
\usepackage{amssymb}%
\usepackage{mathtools}

\begin{document}

\begin{subequations}
\begin{alignat}{3}
  \min &\colon & \quad \alpha & \sum_{i=1}^N\sum_{k=1}^K\sum_{l=1}^K b_{k,l}Y_{i,k}Y_{i,l} + \beta & & \sum_{i=1}^N\sum_{j=1}^M c_{i,j}X_{i,j} + \gamma(\sum_{j=1}^M S_j + T_j) \tag{1} \\
  \text{s.t.} &\phantom{:} & & \sum_{i=1}^N X_{i,j} + S_j - T_j = 1 & & j=1,\ldots,M \\
                 & & & S_j \in \mathbb{Z}^+ \quad && j = 1, \ldots, M \\
                   & & & T_j \in \mathbb{Z}^+ \quad && j = 1, \ldots, M
\end{alignat}
\end{subequations}

\end{document}