Commit 83003845 authored by Michael Keller's avatar Michael Keller
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Initial conclusion

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......@@ -46,34 +46,34 @@ int main()
// int x = 5;
// int y = 2;
// int crops = 10;
// double R[10][10] = {
// {-1, 0, 0.5, -0.5, 1, 0, 0.5, 1, 0.5, 0},
// {0, -1, 0, 1, 0, 0, 0.5, 0.5, 0.5, 0},
// {0.5, 0, -1, 0, 0, 0, 0.5, 0.5, 0, -0.5},
// {-0.5, 1, 0, -1, 0, 0, -1, 0.5, 0.5, 0.5},
// {0, 0, 0, 0, -1, 0, 0, 0, 0, 0},
// {0, 0, 0, 0, 0, -1, 1, 0.5, 0, 0},
// {0.5, 0.5, 0.5, -0.5, 0, 0.5, -1, 0.5, 0.5, 0.5},
// {0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
// {0.5, 0.5, 0, 1, 0, 0, 0.5, 0.5, -1, 0},
// {0, 0, -0.5, 0.5, 0, 0, 1, 0.5, 0.5, -1}};
double R[10][10] = {
{-1, 0, 0.5, -0.5, 1, 0, 0.5, 1, 0.5, 0},
{0, -1, 0, 1, 0, 0, 0.5, 0.5, 0.5, 0},
{0.5, 0, -1, 0, 0, 0, 0.5, 0.5, 0, -0.5},
{-0.5, 1, 0, -1, 0, 0, -1, 0.5, 0.5, 0.5},
{0, 0, 0, 0, -1, 0, 0, 0, 0, 0},
{0, 0, 0, 0, 0, -1, 1, 0.5, 0, 0},
{0.5, 0.5, 0.5, -0.5, 0, 0.5, -1, 0.5, 0.5, 0.5},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{0.5, 0.5, 0, 1, 0, 0, 0.5, 0.5, -1, 0},
{0, 0, -0.5, 0.5, 0, 0, 1, 0.5, 0.5, -1}};
// int distances[] = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1};
char *name = "Arnold (CAUTION, MATRIX DIFFERENT)";
int x = 15;
int y = 15;
int crops = 10;
double R[10][10] = {
{-1, 0, 0.5, -0.5, 0, 1, 0.5, 1, 0, 0.5},
{0, -1, 0, 1, 0, 0, 0.5, 0.5, 0, 0.5},
{0.5, 0, -1, 0, 0, 0, 0.5, 0.5, -0.5, 0},
{-0.5, 1, 0, -1, 0, 0, -1, 0.5, 0.5, 0.5},
{0, 0, 0, 0, -1, 0, 1, 0.5, 0, 0},
{0, 0, 0, 0, 0, -1, 0, 0, 0, 0},
{0.5, 0.5, 0.5, -0.5, 0.5, 0, -1, 0.5, 0.5, 0.5},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{0, 0, -0.5, 0.5, 0, 0, 1, 0.5, -1, 0.5},
{0.5, 0.5, 0, 1, 0, 0, 0.5, 0.5, 0, -1}};
// double R[10][10] = {
// {-1, 0, 0.5, -0.5, 0, 1, 0.5, 1, 0, 0.5},
// {0, -1, 0, 1, 0, 0, 0.5, 0.5, 0, 0.5},
// {0.5, 0, -1, 0, 0, 0, 0.5, 0.5, -0.5, 0},
// {-0.5, 1, 0, -1, 0, 0, -1, 0.5, 0.5, 0.5},
// {0, 0, 0, 0, -1, 0, 1, 0.5, 0, 0},
// {0, 0, 0, 0, 0, -1, 0, 0, 0, 0},
// {0.5, 0.5, 0.5, -0.5, 0.5, 0, -1, 0.5, 0.5, 0.5},
// {0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
// {0, 0, -0.5, 0.5, 0, 0, 1, 0.5, -1, 0.5},
// {0.5, 0.5, 0, 1, 0, 0, 0.5, 0.5, 0, -1}};
int distances[] = {23, 23, 23, 23, 23, 22, 22, 22, 22, 22};
// make the datatypes work
......
\chapter{Calculations Appendix}
\chapter{Appendix}
You can defer lengthy calculations that would otherwise only interrupt
the flow of your thesis to an appendix.
\section{Agroscope Data}
The first matrix is:
\FloatBarrier
\begin{figure}[h]
\centering
\begin{tabular}{ c c c c c c c c c c c }
& $c_1$ & $c_2$ & $c_3$ & $c_4$ & $c_5$ & $c_6$ & $c_7$ & $c_8$ & $c_9$ & $c_{10}$\\
$c_1$ & $-1$ & $0$ & $0.5$ & $-0.5$ & $1$ & $0$ & $0.5$ & $1$ & $0.5$ & $0$\\
$c_2$ & $0$ & $-1$ & $0$ & $1$ & $0$ & $0$ & $0.5$ & $0.5$ & $0.5$ & $0$\\
$c_3$ & $0.5$ & $0$ & $-1$ & $0$ & $0$ & $0$ & $0.5$ & $0.5$ & $0$ & $-0.5$\\
$c_4$ & $-0.5$ & $1$ & $0$ & $-1$ & $0$ & $0$ & $-1$ & $0.5$ & $0.5$ & $0.5$\\
$c_5$ & $0$ & $0$ & $0$ & $0$ & $-1$ & $0$ & $0$ & $0$ & $0$ & $0$\\
$c_6$ & $0$ & $0$ & $0$ & $0$ & $0$ & $-1$ & $1$ & $0.5$ & $0$ & $0$\\
$c_7$ & $0.5$ & $0.5$ & $0.5$ & $-0.5$ & $0$ & $0.5$ & $-1$ & $0.5$ & $0.5$ & $0.5$\\
$c_8$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
$c_9$ & $0.5$ & $0.5$ & $0$ & $1$ & $0$ & $0$ & $0.5$ & $0.5$ & $-1$ & $0$\\
$c_{10}$ & $0$ & $0$ & $-0.5$ & $0.5$ & $0$ & $0$ & $1$ & $0.5$ & $0.5$ & $-1$\\
\end{tabular}
\end{figure}
\FloatBarrier
where
\FloatBarrier
\begin{figure}[h]
\centering
\begin{tabular}{@{}llr@{}} \toprule
Crop Id & Crop Name & Multiplicity $(D(c) \cdot 15^2)$\\ \midrule
$c_1$ & Weisskohl & 23 \\
$c_2$ & Karotten & 23 \\
$c_3$ & Kartoffeln & 23 \\
$c_4$ & Zwiebeln & 23 \\
$c_5$ & Hanf & 23 \\
$c_6$ & Zuckermais & 22 \\
$c_7$ & Buschbohnen & 22 \\
$c_8$ & Kräutermischung & 22 \\
$c_9$ & Kopfsalat & 22 \\
$c_{10}$ & Randen & 22 \\ \bottomrule
\end{tabular}
\end{figure}
\FloatBarrier
and the second matrix is:
\FloatBarrier
\begin{figure}[h]
\centering
\begin{tabular}{ c c c c c c c c c c c c }
& $c_1$ & $c_2$ & $c_3$ & $c_4$ & $c_5$ & $c_6$ & $c_7$ & $c_8$ & $c_9$ & $c_{10}$ & $c_{11}$ \\
$c_1$ & $-0.5$ & $1$ & $1$ & $1$ & $1$ & $1$ & $0.5$ & $-1$ & $0.5$ & $0.5$ & $1$ \\
$c_2$ & $1$ & $-0.5$ & $-1$ & $1$ & $1$ & $0.5$ & $0.5$ & $-1$ & $1$ & $1$ & $1$ \\
$c_3$ & $1$ & $-1$ & $-0.5$ & $-1$ & $1$ & $0$ & $0.5$ & $0.5$ & $0$ & $0$ & $0$ \\
$c_4$ & $1$ & $1$ & $-1$ & $-0.5$ & $0$ & $0$ & $0.5$ & $-1$ & $1$ & $-1$ & $0$ \\
$c_5$ & $1$ & $1$ & $1$ & $0$ & $-0.5$ & $0$ & $0.5$ & $1$ & $1$ & $1$ & $1$ \\
$c_6$ & $1$ & $0.5$ & $0$ & $0$ & $0$ & $-0.5$ & $0.5$ & $0$ & $0$ & $0$ & $0$ \\
$c_7$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ \\
$c_8$ & $-1$ & $-1$ & $0.5$ & $-1$ & $1$ & $0$ & $0.5$ & $-0.5$ & $0$ & $0.5$ & $0$ \\
$c_9$ & $0.5$ & $1$ & $0$ & $1$ & $1$ & $0$ & $0.5$ & $0$ & $-0.5$ & $1$ & $0$ \\
$c_{10}$ & $0.5$ & $1$ & $0$ & $-1$ & $1$ & $0$ & $0.5$ & $0.5$ & $1$ & $-0.5$ & $1$\\
$c_{11}$ & $1$ & $1$ & $0$ & $0$ & $1$ & $0$ & $0.5$ & $0$ & $0$ & $1$ & $-0.5$ \\
\end{tabular}
\end{figure}
\FloatBarrier
where
\FloatBarrier
\begin{figure}[h]
\centering
\begin{tabular}{@{}llr@{}} \toprule
Crop Id & Crop Name & Multiplicity $(D(c) \cdot 15^2)$\\ \midrule
$c_1$ & Kreuzblütler & 20\\
$c_2$ & Schmetterlingsblütler & 20\\
$c_3$ & Nachtschattengewächse & 20\\
$c_4$ & Kartoffel & 20\\
$c_5$ & Korbblütler & 20\\
$c_6$ & Getreide & 20\\
$c_7$ & Kräuter & 21\\
$c_8$ & Liliengewächse & 21\\
$c_9$ & Süssgräser & 21\\
$c_{10}$ & Kürbisgewächse & 21\\
$c_{11}$ & Gänsefussgewächse & 21\\ \bottomrule
\end{tabular}
\end{figure}
\FloatBarrier
\ No newline at end of file
\chapter{Benchmark Problems}
TODO: Add arnold scores
Our goal is to develop methods that work well in
many or all cases. However, data that is based
on real observations is of particular interest.
......@@ -10,85 +8,28 @@ for agriculture, nutrition and the environment,
has compiled some real world data. They have
kindly supplied us with the relationship matrices
so we can see how well our methods work on real data.
The specific matrices can be found in the appendix.
There are two datasets. The first one in general
being more difficult to optimize for than the second.
Throughout this thesis we will report for the presented
methods how well they work on these two matrices. Our
standard field size will be $15 \times 15$ pixels.
The reason for this field size is that that is the size
the team at Agroscope plans on actually testing this
idea out on.
For reference we state here compactly their best known
bounds as well as the scores Dr. Arnold's methods
achieve:
The first matrix is:
\FloatBarrier
\begin{figure}[h]
\centering
\begin{tabular}{ c c c c c c c c c c c c }
& $c_1$ & $c_2$ & $c_3$ & $c_4$ & $c_5$ & $c_6$ & $c_7$ & $c_8$ & $c_9$ & $c_{10}$ & $c_{11}$ \\
$c_1$ & $-0.5$ & $1$ & $1$ & $1$ & $1$ & $1$ & $0.5$ & $-1$ & $0.5$ & $0.5$ & $1$ \\
$c_2$ & $1$ & $-0.5$ & $-1$ & $1$ & $1$ & $0.5$ & $0.5$ & $-1$ & $1$ & $1$ & $1$ \\
$c_3$ & $1$ & $-1$ & $-0.5$ & $-1$ & $1$ & $0$ & $0.5$ & $0.5$ & $0$ & $0$ & $0$ \\
$c_4$ & $1$ & $1$ & $-1$ & $-0.5$ & $0$ & $0$ & $0.5$ & $-1$ & $1$ & $-1$ & $0$ \\
$c_5$ & $1$ & $1$ & $1$ & $0$ & $-0.5$ & $0$ & $0.5$ & $1$ & $1$ & $1$ & $1$ \\
$c_6$ & $1$ & $0.5$ & $0$ & $0$ & $0$ & $-0.5$ & $0.5$ & $0$ & $0$ & $0$ & $0$ \\
$c_7$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ \\
$c_8$ & $-1$ & $-1$ & $0.5$ & $-1$ & $1$ & $0$ & $0.5$ & $-0.5$ & $0$ & $0.5$ & $0$ \\
$c_9$ & $0.5$ & $1$ & $0$ & $1$ & $1$ & $0$ & $0.5$ & $0$ & $-0.5$ & $1$ & $0$ \\
$c_{10}$ & $0.5$ & $1$ & $0$ & $-1$ & $1$ & $0$ & $0.5$ & $0.5$ & $1$ & $-0.5$ & $1$\\
$c_{11}$ & $1$ & $1$ & $0$ & $0$ & $1$ & $0$ & $0.5$ & $0$ & $0$ & $1$ & $-0.5$ \\
\end{tabular}
\end{figure}
\FloatBarrier
where
\FloatBarrier
\begin{figure}[h]
\centering
\begin{tabular}{@{}llr@{}} \toprule
Crop Id & Crop Name\\ \midrule
$c_1$ & Kreuzblütler \\
$c_2$ & Schmetterlingsblütler \\
$c_3$ & Nachtschattengewächse \\
$c_4$ & Kartoffel \\
$c_5$ & Korbblütler \\
$c_6$ & Getreide \\
$c_7$ & Kräuter \\
$c_8$ & Liliengewächse \\
$c_9$ & Süssgräser \\
$c_{10}$ & Kürbisgewächse \\
$c_{11}$ & Gänsefussgewächse \\ \bottomrule
\end{tabular}
\end{figure}
\FloatBarrier
and the second matrix is:
\FloatBarrier
\begin{figure}[h]
\centering
\begin{tabular}{ c c c c c c c c c c c }
& $c_1$ & $c_2$ & $c_3$ & $c_4$ & $c_5$ & $c_6$ & $c_7$ & $c_8$ & $c_9$ & $c_{10}$\\
$c_1$ & $-1$ & $0$ & $0.5$ & $-0.5$ & $1$ & $0$ & $0.5$ & $1$ & $0.5$ & $0$\\
$c_2$ & $0$ & $-1$ & $0$ & $1$ & $0$ & $0$ & $0.5$ & $0.5$ & $0.5$ & $0$\\
$c_3$ & $0.5$ & $0$ & $-1$ & $0$ & $0$ & $0$ & $0.5$ & $0.5$ & $0$ & $-0.5$\\
$c_4$ & $-0.5$ & $1$ & $0$ & $-1$ & $0$ & $0$ & $-1$ & $0.5$ & $0.5$ & $0.5$\\
$c_5$ & $0$ & $0$ & $0$ & $0$ & $-1$ & $0$ & $0$ & $0$ & $0$ & $0$\\
$c_6$ & $0$ & $0$ & $0$ & $0$ & $0$ & $-1$ & $1$ & $0.5$ & $0$ & $0$\\
$c_7$ & $0.5$ & $0.5$ & $0.5$ & $-0.5$ & $0$ & $0.5$ & $-1$ & $0.5$ & $0.5$ & $0.5$\\
$c_8$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
$c_9$ & $0.5$ & $0.5$ & $0$ & $1$ & $0$ & $0$ & $0.5$ & $0.5$ & $-1$ & $0$\\
$c_{10}$ & $0$ & $0$ & $-0.5$ & $0.5$ & $0$ & $0$ & $1$ & $0.5$ & $0.5$ & $-1$\\
\end{tabular}
\end{figure}
\FloatBarrier
where
\FloatBarrier
\begin{figure}[h]
\centering
\begin{tabular}{@{}llr@{}} \toprule
Crop Id & Crop Name\\ \midrule
$c_1$ & Weisskohl \\
$c_2$ & Karotten \\
$c_3$ & Kartoffeln \\
$c_4$ & Zwiebeln \\
$c_5$ & Hanf \\
$c_6$ & Zuckermais \\
$c_7$ & Buschbohnen \\
$c_8$ & Kräutermischung \\
$c_9$ & Kopfsalat \\
$c_{10}$ & Randen \\ \bottomrule
\begin{tabular}{@{}cccc@{}} \toprule
Dataset & Best known bound & Arnold score\\ \midrule
1 & $950$ & $635$ \\
2 & $1624$ & TODO \\
\bottomrule
\end{tabular}
\end{figure}
\FloatBarrier
Throughout this thesis we will report for the presented
methods how well they work on these two matrices. Our
standard field size will be $15 \times 15$ pixels.
\ No newline at end of file
\FloatBarrier
\ No newline at end of file
......@@ -165,5 +165,5 @@ on the shape of the field, or indeed that this score
is achievable with only one field (and not multiple disjoint
fields).
The bound for the first benchmark problem is $1624$, for
the second benchmark problem $950$.
\ No newline at end of file
The bound for the first benchmark problem is $950$, for
the second benchmark problem $1624$.
\ No newline at end of file
\chapter{Conclusion}
\ No newline at end of file
\chapter{Conclusion}
Let us finally look at all of the presented
methods and results together. We first
were able to show that the general
problem is NP complete. This means
that it is most likely computationally
infeasible to solve the problem for
large field with many different types
of crops.
Despite the
general computational difficulty, we
found a method that allows for generating
solutions for large fields from small
ones without worsening the score relative
to the size of the field. This means
we can get a good starting point
for solving large fields by first optimally
solving the small field and then stretching it
to be larger.
Methods for making fractional
solutions into mostly integer solutions
enabled us to try gradient ascent on the
fractional version of pixel farming.
In practice however there were too many local
maxima such that this method, even with
many reruns, did not yield improvements
on the best known scores.
Finally we generalized on Dr. Arnold's idea
of simulated annealing and hill climbing
by using a linear program to find the optimal
swap between multiple pixels instead of
only two. This yielded a slight improvement
in the best known score on our benchmark
problems.
\ No newline at end of file
......@@ -103,4 +103,17 @@ Using these elements we now build the following algorithm:
\Return $integerize(x_0)$
\end{algorithmic}
\end{algorithm}
\FloatBarrier
\ No newline at end of file
\FloatBarrier
Which yields the following scores:
\FloatBarrier
\begin{figure}[h]
\centering
\begin{tabular}{@{}cccc@{}} \toprule
Dataset & Best known bound & Arnold score & Gradient Ascent\\ \midrule
1 & $950$ & $635$ & TODO\\
2 & $1624$ & TODO & TODO\\
\bottomrule
\end{tabular}
\end{figure}
\FloatBarrier
TODO
\ No newline at end of file
\chapter{Computational Hardness}
\chapter{Computational Difficulty}
We show that the problem of finding a
solution of a certain quality to a pixel
......
......@@ -300,8 +300,8 @@ statement with a condition on $R$ might be helpful:
\node[main node, fill=black!20,] (11) [below of=10] {N};
\node[main node, fill=black!20,] (12) [below of=11] {N};
\path[every node/.style={font=\sffamily\small}];
(1) edge[bend left] node [above] {$c_\alpha$} (2);
\path[every node/.style={font=\sffamily\small}]
(1) edge[bend left] node [above] {$c_\alpha$} (2)
(2) edge[bend left] node [below] {$c_\beta$} (1);
\end{tikzpicture}
\end{figure}
......
......@@ -245,4 +245,17 @@ procedure:
\FloatBarrier
After running this algorithm repeatedly for around 24h
on a standard Laptop we were able to generate the
following solutions for our benchmark problems:
\ No newline at end of file
following solutions for our benchmark problems:
\FloatBarrier
\begin{figure}[h]
\centering
\begin{tabular}{@{}cccc@{}} \toprule
Dataset & Best known bound & Arnold score & LP Method\\ \midrule
1 & $950$ & $635$ & $658$\\
2 & $1624$ & TODO & TODO\\
\bottomrule
\end{tabular}
\end{figure}
\FloatBarrier
which are a slight improvement over the previous
best scores.
\ No newline at end of file
......@@ -4,7 +4,7 @@
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......@@ -25,5 +25,6 @@
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\BOOKMARK [0][-]{chapter.10}{\376\377\000C\000o\000n\000c\000l\000u\000s\000i\000o\000n}{}% 27
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\contentsline {proof}{{Proof}{1}{}}{7}{proof.1}%
\contentsline {theorem}{{Theorem}{4.{2}}{}}{9}{theorem.4.3.2}%
\contentsline {proof}{{Proof}{2}{}}{9}{proof.2}%
\contentsline {theorem}{{Theorem}{7.{1}}{}}{18}{theorem.7.1.1}%
\contentsline {proof}{{Proof}{3}{}}{18}{proof.3}%
\contentsline {theorem}{{Theorem}{7.{2}}{}}{20}{theorem.7.2.2}%
\contentsline {proof}{{Proof}{4}{}}{20}{proof.4}%
\contentsline {theorem}{{Theorem}{7.{3}}{}}{21}{theorem.7.2.3}%
\contentsline {proof}{{Proof}{5}{}}{21}{proof.5}%
\contentsline {theorem}{{Theorem}{8.{1}}{}}{27}{theorem.8.1.1}%
\contentsline {proof}{{Proof}{6}{}}{27}{proof.6}%
\contentsline {theorem}{{Theorem}{8.{2}}{}}{27}{theorem.8.1.2}%
\contentsline {proof}{{Proof}{7}{}}{27}{proof.7}%
\contentsline {theorem}{{Theorem}{7.{1}}{}}{16}{theorem.7.1.1}%
\contentsline {proof}{{Proof}{3}{}}{16}{proof.3}%
\contentsline {theorem}{{Theorem}{7.{2}}{}}{18}{theorem.7.2.2}%
\contentsline {proof}{{Proof}{4}{}}{18}{proof.4}%
\contentsline {theorem}{{Theorem}{7.{3}}{}}{19}{theorem.7.2.3}%
\contentsline {proof}{{Proof}{5}{}}{19}{proof.5}%
\contentsline {theorem}{{Theorem}{8.{1}}{}}{25}{theorem.8.1.1}%
\contentsline {proof}{{Proof}{6}{}}{25}{proof.6}%
\contentsline {theorem}{{Theorem}{8.{2}}{}}{25}{theorem.8.1.2}%
\contentsline {proof}{{Proof}{7}{}}{25}{proof.7}%
......@@ -6,28 +6,29 @@
\contentsline {chapter}{\chapternumberline {3}Problem Statement}{5}{chapter.3}%
\contentsline {section}{\numberline {3.1}Integer Version}{5}{section.3.1}%
\contentsline {section}{\numberline {3.2}Fractional Version}{6}{section.3.2}%
\contentsline {chapter}{\chapternumberline {4}Computational Hardness}{7}{chapter.4}%
\contentsline {chapter}{\chapternumberline {4}Computational Difficulty}{7}{chapter.4}%
\contentsline {section}{\numberline {4.1}The decision version of pixel farming}{7}{section.4.1}%
\contentsline {section}{\numberline {4.2}Pixel Farming is NP complete}{7}{section.4.2}%
\contentsline {paragraph}{Hamiltonian Paths}{8}{section*.2}%
\contentsline {paragraph}{Reduction}{8}{section*.3}%
\contentsline {section}{\numberline {4.3}Fractional Pixel Farming is NP Complete}{9}{section.4.3}%
\contentsline {chapter}{\chapternumberline {5}Benchmark Problems}{11}{chapter.5}%
\contentsline {chapter}{\chapternumberline {6}Bounds}{15}{chapter.6}%
\contentsline {section}{\numberline {6.1}Basic Bound}{15}{section.6.1}%
\contentsline {section}{\numberline {6.2}LP bound}{15}{section.6.2}%
\contentsline {chapter}{\chapternumberline {7}Helpful Statements}{17}{chapter.7}%
\contentsline {section}{\numberline {7.1}Growing solutions}{17}{section.7.1}%
\contentsline {section}{\numberline {7.2}Creating integer solutions from fractional solutions}{19}{section.7.2}%
\contentsline {subsection}{\numberline {7.2.1}The standard method}{19}{subsection.7.2.1}%
\contentsline {subsection}{\numberline {7.2.2}The advanced method}{21}{subsection.7.2.2}%
\contentsline {chapter}{\chapternumberline {8}The Linear Programming Method}{25}{chapter.8}%
\contentsline {section}{\numberline {8.1}Problem Setup}{25}{section.8.1}%
\contentsline {section}{\numberline {8.2}Method}{28}{section.8.2}%
\contentsline {chapter}{\chapternumberline {9}Gradient Ascent}{31}{chapter.9}%
\contentsline {section}{\numberline {9.1}Finding the closest valid solution}{31}{section.9.1}%
\contentsline {section}{\numberline {9.2}Computing the gradient}{32}{section.9.2}%
\contentsline {section}{\numberline {9.3}Method}{32}{section.9.3}%
\contentsline {chapter}{\chapternumberline {6}Bounds}{13}{chapter.6}%
\contentsline {section}{\numberline {6.1}Basic Bound}{13}{section.6.1}%
\contentsline {section}{\numberline {6.2}LP bound}{13}{section.6.2}%
\contentsline {chapter}{\chapternumberline {7}Helpful Statements}{15}{chapter.7}%
\contentsline {section}{\numberline {7.1}Growing solutions}{15}{section.7.1}%
\contentsline {section}{\numberline {7.2}Creating integer solutions from fractional solutions}{17}{section.7.2}%
\contentsline {subsection}{\numberline {7.2.1}The standard method}{17}{subsection.7.2.1}%
\contentsline {subsection}{\numberline {7.2.2}The advanced method}{19}{subsection.7.2.2}%
\contentsline {chapter}{\chapternumberline {8}The Linear Programming Method}{23}{chapter.8}%
\contentsline {section}{\numberline {8.1}Problem Setup}{23}{section.8.1}%
\contentsline {section}{\numberline {8.2}Method}{26}{section.8.2}%
\contentsline {chapter}{\chapternumberline {9}Gradient Ascent}{29}{chapter.9}%
\contentsline {section}{\numberline {9.1}Finding the closest valid solution}{29}{section.9.1}%
\contentsline {section}{\numberline {9.2}Computing the gradient}{30}{section.9.2}%
\contentsline {section}{\numberline {9.3}Method}{30}{section.9.3}%
\contentsline {chapter}{\chapternumberline {10}Conclusion}{33}{chapter.10}%
\contentsline {appendix}{\chapternumberline {A}Calculations Appendix}{35}{appendix.A}%
\contentsline {chapter}{Bibliography}{37}{appendix*.4}%
\contentsline {appendix}{\chapternumberline {A}Appendix}{35}{appendix.A}%
\contentsline {section}{\numberline {A.1}Agroscope Data}{35}{section.A.1}%
\contentsline {chapter}{Bibliography}{39}{appendix*.4}%
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