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Commit 7f394000 authored by Dominic Noltekuhlmann's avatar Dominic Noltekuhlmann
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......@@ -111,9 +111,9 @@ Paderborn University}
%%%pythonmarker
% Remove commenting:
%\renewcommand*{\todo}[1]{}
\renewcommand*{\todo}[1]{}
\renewcommand*{\domse}[1]{}
%\renewcommand*{\eli}[1]{}
\renewcommand*{\eli}[1]{}
\begin{document}
......@@ -239,7 +239,7 @@ Paderborn University}
\item subject to $\sum_{i=1}^{n} c_i \cdot x_i \leq C_{max}$ (constraint)
\end{itemize}
\end{itemize}
\item Additionally, we can define the excess cost $e(x):=\sum_{i=1}^{n} c_i \cdot x_i - C_{max}$ used for calculations later.
\item Additionally, we can define the excess cost $e(x):= \left(\sum_{i=1}^{n} c_i \cdot x_i\right) - C_{max}$ used for calculations later.
\end{itemize}
\end{frame}
......@@ -328,7 +328,7 @@ Paderborn University}
\item Three methods:
\begin{itemize}
\item \textbf{Extinctive Penalties}: Brutal penalties.
\item \textbf{Binary Penalties}: Either apply or apply no penalty.
\item \textbf{Binary Penalties}: Either apply a penalty or don't apply a penalty.
\end{itemize}
\end{itemize}
\end{frame}
......@@ -341,7 +341,7 @@ Paderborn University}
\item Three methods:
\begin{itemize}
\item \textbf{Extinctive Penalties}: Brutal penalties.
\item \textbf{Binary Penalties}: Either apply or apply no penalty.
\item \textbf{Binary Penalties}: Either apply a penalty or don't apply a penalty.
\item \textbf{Distance Based Penalties}: Penalty reflects difficulty of being feasible. \domse{weiß noch nicht ganz wie ich das im unterschied zu death penalty erklären soll}
\end{itemize}
\end{itemize}
......@@ -355,7 +355,7 @@ Paderborn University}
\item Three methods:
\begin{itemize}
\item \textbf{Extinctive Penalties}: Brutal penalties.
\item \textbf{Binary Penalties}: Either apply or apply no penalty.
\item \textbf{Binary Penalties}: Either apply a penalty or don't apply a penalty.
\item \textbf{Distance Based Penalties}: Penalty reflects difficulty of being feasible. \domse{weiß noch nicht ganz wie ich das im unterschied zu death penalty erklären soll}
\end{itemize}
\item \textbf{Advantages:} Mostly easy and simple to use tool.\domse{weil dafür gesorgt wird das unmögliche lösungen eliminiert/stark bestraft werden}
......@@ -370,7 +370,7 @@ Paderborn University}
\item Three methods:
\begin{itemize}
\item \textbf{Extinctive Penalties}: Brutal penalties.
\item \textbf{Binary Penalties}: Either apply or apply no penalty.
\item \textbf{Binary Penalties}: Either apply a penalty or don't apply a penalty.
\item \textbf{Distance Based Penalties}: Penalty reflects difficulty of being feasible. \domse{weiß noch nicht ganz wie ich das im unterschied zu death penalty erklären soll}
\end{itemize}
\item \textbf{Advantages:} Mostly easy and simple to use tool.\domse{weil dafür gesorgt wird das unmögliche lösungen eliminiert/stark bestraft werden}
......@@ -623,7 +623,7 @@ Paderborn University}
\includegraphics[width=0.3\textwidth]{../MALEO-report/figures/repair_mapping.pdf}
\begin{itemize}
\item Uses two population sets $P_s$ and $P_r$, for evaluation and repairing.
\item Infeasible solutions are repaired by sampling line segments to feasible points. \domse{taken from a different example but a similar repair process can be observed}
\item Best feasible solution is $z$.
\end{itemize}
\end{frame}
......@@ -638,112 +638,11 @@ Paderborn University}
\begin{itemize}
\item Problem dependent and sometimes complex. \domse{für jedes problem eine spezifische repair function is nötig. in manchen fälle kann das definieren einer solchen funktion genauso schwer sein wie das lösen des problems selbst}
\item Repair functions can be computationally expensive.
\item Repair functions can introduce noise.
\end{itemize}
\end{itemize}
\end{frame}
\subsection{Decoder Functions}
\begin{frame}
\frametitle{Decoder Functions}
\begin{itemize}
\item Maps genotype to phenotype forcing feasibility.
\item Requirements:
\begin{itemize}
\item Every genotype solution must map to a feasible solution.
\item Every feasible solution must have at least one genotype equivalent.
\item Every feasible solution must have the same number of genotype solutions.
\end{itemize}
\item Additional optional requirements:
\begin{itemize}
\item The mapping should be computationally fast.
\item Supports locality feature. \domse{kleine veränderung im genotype region führt zu kleinen änderungen in der möglichen lösung}
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Example: Knapsack Problem (Decoder Function)}
\begin{itemize}
\item Idea: Use a mapping to directly map form the genotype to the feasible solution space $F$.
\item Approach:
\begin{itemize}
\item Step 1. Set $x$ to include no elements. \domse{all $0$ in $x$}
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Example: Knapsack Problem (Decoder Function)}
\begin{itemize}
\item Idea: Use a mapping to directly map form the genotype to the feasible solution space $F$.
\item Approach:
\begin{itemize}
\item Step 1. Set $x$ to include no elements. \domse{all $0$ in $x$}
\item Step 2. Randomly flip a bit from $0$ to $1$.
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Example: Knapsack Problem (Decoder Function)}
\begin{itemize}
\item Idea: Use a mapping to directly map form the genotype to the feasible solution space $F$.
\item Approach:
\begin{itemize}
\item Step 1. Set $x$ to include no elements. \domse{all $0$ in $x$}
\item Step 2. Randomly flip a bit from $0$ to $1$.
\item Step 3. Check whether the last flipped bit causes $e(x) > 0$.
\begin{itemize}
\item If so: return to the previous version of $x$ and terminate.
\item If not: repeat from step 2.
\end{itemize}
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Example: Knapsack Problem (Decoder Function)}
\begin{itemize}
\item Idea: Use a mapping to directly map form the genotype to the feasible solution space $F$.
\item Approach:
\begin{itemize}
\item Step 1. Set $x$ to include no elements. \domse{all $0$ in $x$}
\item Step 2. Randomly flip a bit from $0$ to $1$.
\item Step 3. Check whether the last flipped bit causes $e(x) > 0$.
\begin{itemize}
\item If so: return to the previous version of $x$ and terminate.
\item If not: repeat from step 2.
\end{itemize}
\end{itemize}
\item Fitness of binary solution $x$: $f_D(x) = f(x)$
\end{itemize}
\end{frame}
\begin{frame}{Example: Knapsack Problem (Decoder Function)}
\centering
\includegraphics[width=0.7\textwidth]{../MALEO-report/figures/decoder_mapping.pdf}
\begin{itemize}
\item Mapping $T$ between the genotype solution space $(b)$ and the feasible solution space $(a)$. \domse{$d$ is equal to the binary solution $x$ and $f(x)$ is equal to $s$. The feasible areas of the solution space are shaded}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Evaluation Decoder Function}
\begin{itemize}
\item \textbf{Advantages:} Relative simple approach of using evolutionary algorithms. \domse{idee ist einfach (zu verstehen und zu implementieren), es wird lediglich ein mapping von genotype zu phenotype benötig}
\item \textbf{Disadvantages:} Causes a lot of redundancy, leading to the loss of potential feasible solutions. \domse{in einer one-to-many mapping kann es passieren durch eine schlechte definition des mappings das unterschiedliche genotype lösungen zu einer phenotype lösung gemapped werden, wodurch nicht mehr alle abbildbar sind}
\end{itemize}
\end{frame}
% \begin{frame}{Example: Knapsack Problem (2)}
% \begin{itemize}
% \item The GENOCOP III by Michalewicz’s: \eli{trotzdem möglich das für das knapsack problem etwas abgeändert zu nutzen? oder lieber doch ein ganz anderes bild nutzen?}
......@@ -918,7 +817,8 @@ Paderborn University}
\begin{frame}
\frametitle{Binary Penalties (Static Penalty Function)}
\begin{itemize}
\item General Approach: Replace the excess cost function (distance metric $d^K(x)$) with a binary function $\delta$. \item Form (for Knapsack problem):
\item General Approach: Replace the excess cost function with a binary function $\delta$.
\item Form (for Knapsack problem):
\begin{displaymath}
f_P(x) = f(x) - w \cdot \delta
\text{\qquad where }
......@@ -936,7 +836,7 @@ Paderborn University}
\begin{frame}
\frametitle{Distance Based Penalties (Static Penalty Function)}
\begin{itemize}
\item General Approach: Focus on distance to feasibility, using the excess cost function (distance metric $d^K(\boldsymbol{x})$).
\item General Approach: Focus on distance to feasibility, using the excess cost function.
\item Form (for Knapsack problem):
\begin{displaymath}
f_P(x) = f(x) - w \cdot \max(0, e(x))
......@@ -947,6 +847,106 @@ Paderborn University}
\end{frame}
\subsection{Decoder Functions}
\begin{frame}
\frametitle{Decoder Functions}
\begin{itemize}
\item Maps genotype to phenotype forcing feasibility.
\item Requirements:
\begin{itemize}
\item Every genotype solution must map to a feasible solution.
\item Every feasible solution must have at least one genotype equivalent.
\item Every feasible solution must have the same number of genotype solutions.
\end{itemize}
\item Additional optional requirements:
\begin{itemize}
\item The mapping should be computationally fast.
\item Supports locality feature. \domse{kleine veränderung im genotype region führt zu kleinen änderungen in der möglichen lösung}
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Example: Knapsack Problem (Decoder Function)}
\begin{itemize}
\item Idea: Use a mapping to directly map form the genotype to the feasible solution space $F$.
\item Approach:
\begin{itemize}
\item Step 1. Set $x$ to include no elements. \domse{all $0$ in $x$}
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Example: Knapsack Problem (Decoder Function)}
\begin{itemize}
\item Idea: Use a mapping to directly map form the genotype to the feasible solution space $F$.
\item Approach:
\begin{itemize}
\item Step 1. Set $x$ to include no elements. \domse{all $0$ in $x$}
\item Step 2. Randomly flip a bit from $0$ to $1$.
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Example: Knapsack Problem (Decoder Function)}
\begin{itemize}
\item Idea: Use a mapping to directly map form the genotype to the feasible solution space $F$.
\item Approach:
\begin{itemize}
\item Step 1. Set $x$ to include no elements. \domse{all $0$ in $x$}
\item Step 2. Randomly flip a bit from $0$ to $1$.
\item Step 3. Check whether the last flipped bit causes $e(x) > 0$.
\begin{itemize}
\item If so: Return to the previous version of $x$ and terminate.
\item Else: Repeat from step 2.
\end{itemize}
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Example: Knapsack Problem (Decoder Function)}
\begin{itemize}
\item Idea: Use a mapping to directly map form the genotype to the feasible solution space $F$.
\item Approach:
\begin{itemize}
\item Step 1. Set $x$ to include no elements. \domse{all $0$ in $x$}
\item Step 2. Randomly flip a bit from $0$ to $1$.
\item Step 3. Check whether the last flipped bit causes $e(x) > 0$.
\begin{itemize}
\item If so: return to the previous version of $x$ and terminate.
\item If not: repeat from step 2.
\end{itemize}
\end{itemize}
\item Fitness of binary solution $x$: $f_D(x) = f(x)$
\end{itemize}
\end{frame}
\begin{frame}{Example: Knapsack Problem (Decoder Function)}
\centering
\includegraphics[width=0.7\textwidth]{../MALEO-report/figures/decoder_mapping.pdf}
\begin{itemize}
\item Mapping $T$ between the genotype solution space $(b)$ and the feasible solution space $(a)$. \domse{$d$ is equal to the binary solution $x$ and $f(x)$ is equal to $s$. The feasible areas of the solution space are shaded}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Evaluation Decoder Function}
\begin{itemize}
\item \textbf{Advantages:} Relative simple approach of using evolutionary algorithms. \domse{idee ist einfach (zu verstehen und zu implementieren), es wird lediglich ein mapping von genotype zu phenotype benötig}
\item \textbf{Disadvantages:} Causes a lot of redundancy, leading to the loss of potential feasible solutions. \domse{in einer one-to-many mapping kann es passieren durch eine schlechte definition des mappings das unterschiedliche genotype lösungen zu einer phenotype lösung gemapped werden, wodurch nicht mehr alle abbildbar sind}
\end{itemize}
\end{frame}
%\begin{frame}
% \frametitle{Evaluation of Penalty Functions}
% \begin{itemize}
......
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