I've taken some liberties in interpreting what you're trying to accomplish here. For example, I assume that you will manually create the tag names for the LP blocks. Also, I think my solution lacks a certain elegance: in particular, each such block has to be created separately. I don't really like that.
Here's the general idea: create two adjacent boxes centered on the baseline with one containing a dummy equation to create the label for the entire block. Spacing after the boxes is a bit wonky. So, I add ~ after the last flalign environment. (Short coming: you also have to manually adjust the width of the first parbox to get the label to be positioned correctly.)
You seemed to want the first line to somehow align with the LP. I've got two solutions to that. The first puts the first line into a box of zero width. (Short coming: you have to re-enter math mode within this box.) The second puts this first line in its own flalign environment. (Short coming: you have to adjust the vertical spacing between the two environments.)
\documentclass{article}
\usepackage{amsmath,calc}
\newlength{\LPlhbox}
\begin{document}
\settowidth{\LPlhbox}{(P.1)}%
\noindent%
\parbox{\LPlhbox}{\begin{align}
\tag{P.1}\label{My first LP Block}
\end{align}}%
\hspace*{\fill}%
\begin{minipage}{\linewidth-2cm}
\begin{flalign}\notag
& \makebox[0pt][l]{$\displaystyle{}\max \sum\limits_{v \in V} x_v + \sum\limits_{e \in E} y_e + 10000\gamma$} \\
\label{this line can be referenced}
& \text{s.t.} & x_v + x_w & \geq 0 && \forall \{v,w\} \in E && \\
& & y_3 & \geq 0 && \forall e\in E && \\
& & (\delta(U)) & \geq \frac{1}{2}(|U|-1) && \forall U \subseteq V
\end{flalign}~
\end{minipage}
\settowidth{\LPlhbox}{(P.2)}%
\noindent%
\parbox{\LPlhbox}{\begin{align}
\tag{P.2}\label{My second LP Block}
\end{align}}%
\hspace*{\fill}%
\begin{minipage}{\linewidth-2cm}
\begin{flalign}\notag
\max \sum\limits_{v \in V} x_v + \sum\limits_{e \in E} y_e + 10000\gamma &&
\end{flalign}
\vspace{-1.5\baselineskip}
\begin{flalign}
& \text{s.t.} & x_v + x_w & \geq 0 && \forall \{v,w\} \in E && \\
& & y_3 & \geq 0 && \forall e\in E && \\
& & (\delta(U)) & \geq \frac{1}{2}(|U|-1) && \forall U \subseteq V
\end{flalign}~
\end{minipage}
Notice that we can reference line~\ref{this line can be referenced}
and both block~\ref{My first LP Block} and block~\ref{My second LP Block}
\end{document}
What would be nice is if there were a commands \lefttag and \righttag to temporarily override document defaults.
You can also easily create a command to do the work of the left-hand box.
\newcommand{\LPblocktag}[2]{\settowidth{\LPlhbox}{(#1)}%
\parbox{\LPlhbox}{\begin{align}\tag{#1}#2\end{align}}%
\hspace*{\fill}}
I've written the second argument as I have because I don't want to assume you'll be referencing every such block.
So the first block above can be written as
\noindent%
\LPblocktag{P.3}{\label{My third LP Block}}%
\begin{minipage}{\linewidth-2cm}
\begin{flalign}\notag
& \makebox[0pt][l]{$\displaystyle{}\max \sum\limits_{v \in V} x_v + \sum\limits_{e \in E} y_e + 10000\gamma$} \\
\label{this line can be referenced}
& \text{s.t.} & x_v + x_w & \geq 0 && \forall \{v,w\} \in E && \\
& & y_3 & \geq 0 && \forall e\in E && \\
& & (\delta(U)) & \geq \frac{1}{2}(|U|-1) && \forall U \subseteq V
\end{flalign}~
\end{minipage}
I can reference \ref{My third LP Block}
You can number each line using the alignat environment from the amsmath package.
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{alignat}{2}
a &= b &\quad\Rightarrow\quad c &= d \\
e &= f & g &= h
\end{alignat}
\end{document}
You can use \nonumber to suppress numbering on individual lines. I don't think it would be a good idea to separately number the equations in the left-hand column. Readers expecting to find the equation numbers at the edge of the page
are likely to find this annoying.
Best Answer
You can define a custom
nbytwosubequationsenvironment to number subequations two by two, line by line.Caution: this solution assumes that each line contains two subequations; the numbering will be wrong if one line contains only one subequation.