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\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\title{PRODES ILP}
\author{Guilherme Kunigami}
\date{September 2019}
\begin{document}
\maketitle
\section{Introduction}
This document describes an integer linear programming for the Protein Design model PRODES.
\section{Model}
\textbf{Constants.} Let $P$ as the set of positions. Let $R_p$ be the set of rotamers available at position $p \in P$. Let $R$ be the set of all rotamers (i.e. $R = \bigcup_{p in P} R_p$). The coefficient $e_{i,j}$ indicates the cost between rotamer $i$ and $j$.
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\textbf{Variables.} Let $r_{i} \in \{0, 1\}$. $r_{i} = 1$ if we pick rotamer $i \in R$. Let $x_{i,j} \in \{0, 1\}$, $x_{i,j} = 1$ if both rotamer $i$ and $j$ are picked, for all $i \in R_p, j \in R_q$, where $p, q \in P$ and $p < q$.
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\paragraph{Objective function}
$$
\min \sum_{i} \sum_{j, j < i}x_{i,j} e_{i,j}
$$
\paragraph{Constraints}
\begin{align}
\sum_{i \in R_p} r_{i} & = 1 \quad \forall p \in R \\
r_{i} & = \sum_{j \in R_q} x_{i, j} \quad \forall i \in R_p, \mbox{such that } p, q \in P, p < q \\
r_{j} & = \sum_{i \in R_p} x_{i, j} \quad \forall j \in R_q, \mbox{such that } p, q \in P, p < q
\end{align}
Equation (1) says we should pick exactly one rotamer for each position. Constraints (2) and (3) enforces that $x_{i, j} = 1$ if and only if $r_i = r_j = 1$.
\end{document}
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