From 3f5b197643bf6494c50e8667bd7e2542430eb8dd Mon Sep 17 00:00:00 2001
From: msurl <masur101@hhu.de>
Date: Fri, 22 May 2020 01:13:15 +0200
Subject: [PATCH] added first draft of introduction to ilp

---
 Latex/ilp.tex | 21 +++++++++++++++++++++
 1 file changed, 21 insertions(+)
 create mode 100644 Latex/ilp.tex

diff --git a/Latex/ilp.tex b/Latex/ilp.tex
new file mode 100644
index 0000000..cf5be82
--- /dev/null
+++ b/Latex/ilp.tex
@@ -0,0 +1,21 @@
+\section{Linear Programming} \raggedbottom
+Linear programming is a technique to minimize linear functions. 
+A linear programm (LP) consists of a objective function, which is minimized with respect to a set of linear inequalities. \\
+\\
+Linear programms can be expressed as 
+\[min\{c^Tx : Ax \geq b, x \geq 0\}\]
+$b \in \mathbb{R}^n$ and $c \in \mathbb{R}^n$ are constant vectors. $Ax \leq b$ denotes the inequalities which have to be respected. $A \in \mathbb{R}^{m \times n}$ is a matrix containing the coefficients of the $m$ inequalities. $c^Tx \in R$ is the objective function which is minimized. 
+$x \in \mathbb{R}$ is a vector describing possible solutions. If $x \in \mathbb{R}$ obeys all inequalities it is called a feasible solution. A solution $x^*$ is optimal if respects all inequalities and is minimal. 
+\\
+\\
+Integer linear programms (ILPs) are linear programms with the additional restriction that all variables have to be integers: $x \in \mathbb{Z}$. 
+The decision variant of an ILP is NP-complete.
+\\
+In this thesis we use a more readable notation which does not completely sticks to this definition. We write down the inequalities in a sum notation. (Maybe an example?) But it is possible to transform them into the matrix notation. 
+\\
+\\
+Decision problems can be modelles with ILPs. Every variable $x_i \in \{0,1\}$ denotes a possible decision. In our case we assign a decision variable for each vertex to decide if it is included in a dominating set or not. 
+
+
+\pagebreak
+
-- 
GitLab