diff --git a/Main.tex b/Main.tex
index 129ed193c4031714e5b771657e4788e29baa4274..ce76de2a3b47133a392445c7e672580cf09a118e 100644
--- a/Main.tex
+++ b/Main.tex
@@ -204,13 +204,13 @@
\frame{
\frametitle{Supervised Learning}
\begin{columns}[T]
- \begin{column}{0.68\textwidth}
+ \begin{column}{0.6\textwidth}
\phantom{a}\\\medskip
\tikzstyle TestEdge=[-Triangle,line width=3pt,shorten >=3pt,green!40!black,dotted]
\tikzstyle TrainEdge=[-Triangle,thick,shorten >=3pt,line width=3pt]
\tikzset{
Legend/.style={draw=white, top color=white, bottom color=white, inner sep=0.5em, minimum height=1cm,text width=10mm, align = left}}
-\scalebox{0.7}{
+\scalebox{0.72}{
\begin{tikzpicture}[->, node distance=0.8cm, auto]
\small
\node[Legend] (Inst) {};%Trainingsdatenpunkte $i=1 \dots n$
@@ -219,8 +219,8 @@
\node[mynode,below=1.5cm of Inst,align=center] (ML) {ML-Model };
\node[mynode,below=4cm of Inst,align=center] (Pred) {Trained ML-Model};
-\node[mynode,below left =2 cm and 0.5cm of Pred] (TestFeat) {Input $\hat{X}_1,, \dots, \hat{X}_m$};
-\node[mynode,below right=2 cm and 0.5cm of Pred] (OutTest) {Output $Y^*_1,\dots, Y^*_m$};
+\node[mynode,below left =2 cm and 0.5cm of Pred] (TestFeat) {Input $\hat{X}$};
+\node[mynode,below right=2 cm and 0.5cm of Pred] (OutTest) {Output $\hat{Y}$};
\only<2>{
\node[mynodeB,below left=-1cm and 0.5cm of Inst] (Feat) {\textbf{Input $X_1, \dots, X_n$}};%Feature-Vektoren
@@ -228,12 +228,13 @@
}
\only<3>{
\node[mynodeB,below=1.5cm of Inst,align=center] (ML) {\textbf{ML-Model} };
+\node[mynodeB,below=4cm of Inst,align=center] (Pred) {\textbf{Trained ML-Model}};
}
\only<4>{
-\node[mynodeB,below left =2 cm and 0.5cm of Pred] (TestFeat) {Input $\hat{X}_1,, \dots, \hat{X}_m$};
+\node[mynodeB,below left =2 cm and 0.5cm of Pred] (TestFeat) {\textbf{Input $\hat{X}$}};
}
\only<5>{
-\node[mynodeB,below right=2 cm and 0.5cm of Pred] (OutTest) {Output $Y^*_1,\dots, Y^*_m$};
+\node[mynodeB,below right=2 cm and 0.5cm of Pred] (OutTest) {\textbf{Output $\hat{Y}$}};
}
\path
%(TestInst) edge[TestEdge] (TestFeat)
@@ -248,21 +249,21 @@
(Out) edge[TrainEdge] node {}(ML)
(ML) edge[TrainEdge] node {}(Pred)
;
-\node[Legend, below left = 1.5cm and 1.5cm of Inst] (E1) {Training};
-\node[Legend, below left= 2.5cm and 1.5cm of Inst] (E2) {Test};
-\node[Legend, left = 1cm of E1] (S1) {};
-\node[Legend, left = 1cm of E2] (S2) {};
-
-\path
-(S1) edge[TrainEdge] (E1)
-(S2) edge[TestEdge] (E2);
+%\node[Legend, below left = 1.5cm and 1.5cm of Inst] (E1) {Training};
+%\node[Legend, below left= 2.5cm and 1.5cm of Inst] (E2) {Test};
+%\node[Legend, left = 1cm of E1] (S1) {};
+%\node[Legend, left = 1cm of E2] (S2) {};
+
+%\path
+%(S1) edge[TrainEdge] (E1)
+%(S2) edge[TestEdge] (E2);
\end{tikzpicture}
}
\end{column}
- \begin{column}{0.3\textwidth}
+ \begin{column}{0.38\textwidth}
\begin{overprint}
\only<2>{
- \includegraphics[width=\textwidth]{figure/catsdog2}}
+ \includegraphics[width=0.9\textwidth]{figure/catsdog2}}
\only<3>{
\includegraphics[width=\textwidth]{figure/GNN}
@@ -297,20 +298,20 @@
\scriptsize
\node[mynode] at (2,10.5) (1) {Problem};
-\node[mynode2] at (2,9) (2) {OR-expert};
+\node[mynode] at (2,9) (2) {Formalisation};
\node[mynode] at (2,7.5) (3a) {Mathematical \\ model};
\node[mynode] at (2,6) (5a) {Algorithm};
-\visible<4->{
-\node[mynodeB] at (0.5,4) (5b) {\textbf{\normalsize{Data set}}};
+\visible<5->{
+\node[mynodeB] at (0.5,4) (5b) {\textbf{\normalsize{Instance}} \\{\scriptsize(specific data set)}};
\node[mynodeB] at (3.5,4) (6) {\textbf{\normalsize{Solution}}};
}
\draw[edge] (1) -> (2);
\draw[edge] (2) -> (3a);
\draw[edge] (3a) -> (5a);
-\visible<4->{
+\visible<5->{
\draw[edge] (5a) -> (6);
\draw[edge] (5b) -> (5a);
}
@@ -319,10 +320,14 @@
\onslide<1| trans:1>{
\node[mynodeB] at (2,10.5) (a) {\normalsize{ \textbf{Problem}}};
}
-\visible<2| trans:2>{
+\onslide<2| trans:2>{
+\node[mynodeB] at (2,9) (2) {\normalsize{ \textbf{Formalisation}}};
+}
+
+\visible<3| trans:3>{
\node[mynodeB] at (2,7.5) (b) { \textbf{\normalsize{Mathematical}} \\ \textbf{\normalsize{model}}};
}
-\onslide<3| trans:3>{
+\onslide<4| trans:4>{
\node[mynodeB] at (2,6) (c) {\normalsize{\textbf{Algorithm}} };
}
%\onslide<4| trans:4>{
@@ -343,14 +348,19 @@
Problem:\hfill\mbox{}\\[2ex]
\includegraphics[width=.99\textwidth]{figure/ZIMPL1}
\end{bspBlock}
+ \onslide<2| trans:2>
+ \begin{bspBlock}{(Shortest path problem)}
+ Problem:\hfill\mbox{}\\[2ex]
+ \includegraphics[width=.99\textwidth]{figure/ZIMPL0}
+ \end{bspBlock}
%
-\onslide<2| trans:2>
+\onslide<3| trans:3>
\begin{bspBlock}{(Shortest path problem)}
General mathematical model:\hfill\mbox{}\\[2ex]
\includegraphics[width=.99\textwidth]{figure/ZIMPL2}
\end{bspBlock}
%
-\onslide<3| trans:3>
+\onslide<4| trans:4>
\begin{bspBlock}{(Dijkstra's Algorithm)}
\tiny
\begin{algorithm}[H]
@@ -388,9 +398,9 @@
%%
%
%%
-\onslide<4| trans:4 >
+\onslide<5| trans:5 >
\begin{bspBlock}{(Solution)}
- \includegraphics[width=\textwidth]{figure/ZIMPL9b}
+ \includegraphics[width=\textwidth]{figure/spp_}
\end{bspBlock}
\end{overprint}
@@ -400,19 +410,19 @@
\frame{
-\frametitle{Machine Learning and Operations Research}
+\frametitle{Machine Learning(ML) and Operations Research(OR)}
\Large
\textbf{Key Question:}\\
\begin{center}
-\red{How can ML be used for solving optimisation problems?}\\[7ex]\pause
+\red{How can ML be used for solving optimization problems?}\\[7ex]\pause
-Combinatorial optimisation problems are quite different from most problems currently solved by ML \citep{BengioEtAl2021}
+Combinatorial optimization problems are quite different from most problems currently solved by ML \citep{BengioEtAl2021}
\end{center}
}
\section{Vehicle Routing Problems}
\frame{
-\frametitle{The Family of Vehicle Routing Problems}
+\frametitle{Example: Family of Vehicle Routing Problems}
\onslide<1->{
\scalebox{0.95}{\alt<1|trans:0>{\input{figure/BaseVRP_Prob}}{\input{figure/BaseVRP_Routes}}}
\hspace{0.15cm}
@@ -455,7 +465,7 @@ Combinatorial optimisation problems are quite different from most problems curre
\frame{
-\frametitle{The Family of Vehicle Routing Problems}\small
+\frametitle{Family of Vehicle Routing Problems}\small
The family of \textbf{Vehicle Routing Problems (VRPs)} forms one of the most important and most widely studied problems in logistics and combinatorial optimization. They are
\begin{itemize}\small
\item highly \blue{relevant in practice}
@@ -498,16 +508,23 @@ z_{3I-MTZ} &=& \min \sum_{k\in K} \sum_{(i,j) \in A} c_{ij}x^k_{ij}\label{mod:3I
\frame{
\frametitle{Learning Solutions directly?}
\footnotesize
+
Two questions regarding a solution must be answered:\\
-Feasibility (often difficult to evaluate) and Quality (Solution value)
+\blue{Feasibility} (often difficult to evaluate) and \blue{Quality} (Solution value)
\pause
\newcommand{\feasiblecolor}{green!20!white}
\begin{columns}
-\begin{column}{0.15\textwidth}
-Example:
+\begin{column}{0.25\textwidth}
+Example:\\[1ex]
+\visible<4->{
+\green{Optimal Solution}\\[1ex]
+}
+\visible<5->{
+\red{Learned Solution}\\ \red{(Infeasible)}
+}
\end{column}
-\begin{column}{0.85\textwidth}
+\begin{column}{0.7\textwidth}
\begin{tikzpicture}
\node[fill=gray!50,circle,minimum size=5pt,inner sep=0pt] at (4,0) {};
@@ -557,40 +574,45 @@ Example:
\draw[-Latex,thick] (0,0) -- node[pos=1.03] {$x_1$} (0,4);
\draw[-Latex,thick] (0,0) -- node[pos=1.03] {$x_2$} (5,0);
-\node[fill=red,circle,minimum size=5pt,inner sep=0pt] at (0,0) {};
-\node[fill=red,circle,minimum size=5pt,inner sep=0pt] at (1,0) {};
-\node[fill=red,circle,minimum size=5pt,inner sep=0pt] at (2,0) {};
-\node[fill=red,circle,minimum size=5pt,inner sep=0pt] at (3,0) {};
-\node[fill=red,circle,minimum size=5pt,inner sep=0pt] at (0,1) {};
-\node[fill=red,circle,minimum size=5pt,inner sep=0pt] at (1,1) {};
-\node[fill=red,circle,minimum size=5pt,inner sep=0pt] at (2,1) {};
+\node[fill=blue,circle,minimum size=5pt,inner sep=0pt] at (0,0) {};
+\node[fill=blue,circle,minimum size=5pt,inner sep=0pt] at (1,0) {};
+\node[fill=blue,circle,minimum size=5pt,inner sep=0pt] at (2,0) {};
+\node[fill=blue,circle,minimum size=5pt,inner sep=0pt] at (3,0) {};
+\node[fill=blue,circle,minimum size=5pt,inner sep=0pt] at (0,1) {};
+\node[fill=blue,circle,minimum size=5pt,inner sep=0pt] at (1,1) {};
+\node[fill=blue,circle,minimum size=5pt,inner sep=0pt] at (2,1) {};
}
-
\only<4->{
-\node[fill=blue,circle,minimum size=5pt,inner sep=0pt] at (2,2) {};
+\node[fill=green,circle,minimum size=5pt,inner sep=0pt] at (3,0) {};
+}
+
+\only<5->{
+\node[fill=red,circle,minimum size=5pt,inner sep=0pt] at (3,1) {};
}
\end{tikzpicture}
\end{column}
\end{columns}
-\pause\pause
-Good/optimal solutions are often located in the border region.
-\red{$\Rightarrow$}Learning solutions for highly constrained problems seems not promising:
+\pause\pause\phantom{a}\\[0.5ex]
+Good/optimal solutions are often located in the border region.\\[0.5ex]
+\pause
+\red{$\Rightarrow$}Learning solutions for highly constrained problems seems not promising:\\[0.5ex]
\begin{itemize}
-\item Guaranteeing that the learned solution is feasible is rarely possible
-\item No guarantees in terms of solution quality can be given
+\item Guaranteeing feasibility is rarely possible
+\item No guarantees in terms of solution quality
\end{itemize}
}
\frame{
\frametitle{Combine ML with OR Algorithms!}
\small
- Sometimes expert knowledge is not satisfactory and algorithmic decisions are taken greedily or according to ``best practice''.\\\bigskip\pause
+ Status Quo in OR-algorithms:\\
+ Sometimes expert knowledge is not satisfactory and \blue{algorithmic decisions} are taken \blue{greedily} or according to ``\blue{best practice}''.\\\bigskip\pause
{\large
-\blue{$\Rightarrow:$ Apply learning inside OR Algorithms} \citep{BengioEtAl2021}:\medskip}\pause
+\blue{$\Rightarrow:$ Apply learning inside OR Algorithms} :\medskip}\pause
\begin{itemize}
%\item End to end learning (only for little constraint problems like TSP)
@@ -598,8 +620,10 @@ Good/optimal solutions are often located in the border region.
\item Machine learning alongside optimization algorithms (recurring decisions)
\end{itemize}
\bigskip\pause
-Most ML Applications in OR focus on heuristics!
-
+\begin{center}
+{\large
+\green{Most ML Applications in OR focus on heuristics!}\\}\citep{BengioEtAl2021}
+\end{center}
}
@@ -609,7 +633,7 @@ Most ML Applications in OR focus on heuristics!
\includegraphics[scale=0.25]{figure/fwflogo}
\phantom{a}\\\bigskip\phantom{a}\\
{\Large
-Using supervised Learning in Branch-Price-and-Cut for Vehicle Routing Problems
+Using Supervised Learning in Branch-Price-and-Cut for Vehicle Routing Problems
}
\end{center}
\begin{columns}
@@ -669,8 +693,9 @@ z_{3I-MTZ} &=& \min \sum_{k\in K} \sum_{(i,j) \in A} c_{ij}x^k_{ij}\label{mod:3I
\frametitle{Dantzig-Wolfe Reformulation}\small
\begin{columns}[T]
\begin{column}{0.42\textwidth}
+\centering
%\footnotesize
-\mygreen{Aggregated problem formu-\\lation with routing variables:}\\[0.5ex]
+\mygreen{Aggregated problem formu-\\lation with routing variables:}\\[1ex]
\scriptsize
\blue{$\lambda_{r}$}: Binary variable with $\lambda_{r} = 1$ iff a \blue{vehicle travels the route} $r\in \Omega$
\begin{eqnarray*}\setcounter{equation}{2}
@@ -681,11 +706,13 @@ z_{3I-MTZ} &=& \min \sum_{k\in K} \sum_{(i,j) \in A} c_{ij}x^k_{ij}\label{mod:3I
\end{eqnarray*}
\footnotesize
\medskip
-$\cDTo$ \textbf{\red{Set-Partitioning~formulation}}
+Set-Partitioning~formulation
+\red{with exponential many variables}
\end{column}
\begin{column}{0.64\textwidth}
-\visible<3->{\hspace*{-3ex}
-\hspace*{2ex}\green{Definition of routing variables:}
+\centering
+\visible<3->{
+\green{Definition of routing variables:}
\[
\Omega = \left\{ r = (y_i, x_{ij})\right\}\text{, with}
\]
@@ -704,7 +731,8 @@ $\cDTo$ \textbf{\red{Set-Partitioning~formulation}}
\end{column}
\end{columns}
\visible<4->{
- Model can \blue{\underline{not} be fully formulated and solved} using standard software $\Rightarrow$ Solution via \blue{branch-and-price-and cut} }
+\centering
+ Model can \red{\underline{not} be fully formulated and solved} using standard software\\ $\Rightarrow$ Solution via \blue{Branch-Price-and-Cut} }
\end{frame}
\begin{frame}
@@ -818,8 +846,8 @@ $\cDTo$ \textbf{\red{Set-Partitioning~formulation}}
\small
Where can ML help?\\\pause
\begin{itemize}[<+->]
-\item Branching and Cutting decision \citep{FurianEtAl2021}
-\item Column Selection \citep{MorabitEtAl2021}
+\item Branching decisions \citep{FurianEtAl2021}
+\item Column selection \citep{MorabitEtAl2021}
\item Help solving the subproblem \citep{MorabitEtAl2023}
\item \green{Deciding on a relaxation for the subproblem}
\item \dots
@@ -896,10 +924,10 @@ Find $s$-$t$-path with \blue{minimum cost} such that\\
\bigskip\pause
\textbf{SPPRC with $ng$-paths \citep{BaldacciEtAl2011}:}
\begin{itemize}
- \item \blue{Idea:} prohibit/allow certain cycles to handle trade-off between strength of the relaxation and difficulty of the subproblem
- \item A \blue{neighborhood} $N_i\subset V\setminus\{s,t\}$ is associated with each node $i\in V$, e.g.~$i$ and its nearest neighbors
- \item A \blue{cycle with a node $i$} is allowed only if the circle contains a node $j$ with $i\notin N_j$
- \item \blue{Special cases of SPPRC with $ng$-routes}:\\
+ \item \blue{Idea:} \blue{allow only certain cycles}\\ (strength of relaxation vs difficulty of subproblem)
+ \item A \blue{neighborhood} $N_i\subset V\setminus\{s,t\}$ is associated with each node $i\in V$ %, e.g.~$i$ and its nearest neighbors
+ \item \blue{Cycle with node $i$} is allowed if the cycle contains $j$ with $i\notin N_j$
+ \item \blue{Special cases} of SPPRC with $ng$-paths:\\
$\cTo$ $N_i = N$ for all $i$ $\Rightarrow$ \blue{ESPPRC}\\
$\cTo$ $N_i = \{i\}$ for all $i$ $\Rightarrow$ \blue{SPPRC}\\
\end{itemize}
@@ -916,12 +944,12 @@ Find $s$-$t$-path with \blue{minimum cost} such that\\
\frame{
\frametitle{Learning $ng$-path relaxation}\small
-Size of neighborhoods and selection of neighbors have a huge impact on the overall algorithmic performance!\\\smallskip
-Usually: Policy-based neighborhoods (e.g., choose the $x$ closed nodes)\\\bigskip\pause
+Size of \blue{neighborhoods} and selection of neighbors \blue{have a huge impact} on the overall algorithmic performance!\\\medskip\pause
+Status Quo: \blue{Policy-based} neighborhoods\\(e.g., choose the $x$ closed nodes)\\\bigskip\pause\medskip
%
-What is a \blue{good neighborhood}?\\\smallskip\pause
-A \blue{smallest} neighborhood leading to an \blue{elementary solution} of the LP-Relaxation\\\bigskip\pause
-How can we find such a neighborhood?\\\smallskip\pause \blue{Dynamic Neighborhood Extension} \citep[DNE, see][]{BodeIrnich2015,RobertiMingozzi2014}
+What is a \blue{good neighborhood}?\\\medskip\pause
+A \blue{smallest} neighborhood leading to an \blue{elementary solution} of the LP-Relaxation\\\bigskip\pause\medskip
+How can we find such a neighborhood?\\\medskip\pause \blue{Dynamic Neighborhood Extension} \citep[DNE, see][]{BodeIrnich2015,RobertiMingozzi2014}
}
\frame{
@@ -933,12 +961,12 @@ How can we find such a neighborhood?\\\smallskip\pause \blue{Dynamic Neighborhoo
\end{enumerate}
\medskip\pause
\begin{itemize}[<+->]
- \item[\green{$+$}] The neighborhood obtained at the end is a smallest that achieves the so-called elementary lower bound
+ \item[\green{$+$}] The neighborhood obtained at the end is (similar to) a smallest that achieves the so-called elementary lower bound
\item[\red{$-$}] Quite time consuming (Full CG-algorithm in each iteration)
\end{itemize}
\begin{center}
\large
-\medskip\pause \green{Use supervised learning to get such a neighborhood a priori?}
+\medskip\pause \green{Use supervised learning to get such a neighborhood a priori}
\end{center}
}
\begin{frame}{Examples for Neighborhoods obtained by DNE}
@@ -947,7 +975,7 @@ How can we find such a neighborhood?\\\smallskip\pause \blue{Dynamic Neighborhoo
\end{frame}
\section{First results}
-\begin{frame}{Generating data}
+\begin{frame}{Computational Results -- Generating Data}
\small
\begin{itemize}
\item Over 20,000 VRPTW-instances with 100 customers each are generated
@@ -979,7 +1007,7 @@ How can we find such a neighborhood?\\\smallskip\pause \blue{Dynamic Neighborhoo
\item Duration of cycle $i-j-i$ %\red{$b_j-(a_i+t_{ij})$}
\end{itemize}\pause
%\item On top of that, also Graph Encoding variables of the kNN-induced graph can be considered. \red{k=20}
-
+ \medskip
\item The neighborhood obtained with DNE is used as the target for classification tasks.
\end{itemize}
\end{frame}
@@ -995,8 +1023,8 @@ How can we find such a neighborhood?\\\smallskip\pause \blue{Dynamic Neighborhoo
Models used to perform the classification task:\\\medskip
\begin{itemize}
\item[\modA] Deep classification head of GNN encoded variables
- \item[\modB] Homogeneous GNN
- \item[\modC] Heterogeneous GNN \pause
+ \item[\modB] Homogeneous GNN to predict $ng$-graph
+ \item[\modC] Heterogeneous GNN to predict $ng$-graph \pause
\item[\RF] Random Forest with engineered variables
\end{itemize}
@@ -1136,26 +1164,26 @@ Accuracy 0.895
\caption{\RF, Accuracy 0.923}
\end{figure}
\end{frame}
-\begin{frame}{DNE-neighborhood vs learned neighborhood}
-
-
-Picture neighborhood vs learned neighborhood - are the learned rather sparse or have more 1-entries then necessary?
-%\red{Sensitivity means correctly guessed 1's right? Its impotant cause structure is sparse\\
-%\red{In distribution is test set and out distribution is e.g. solomon instances, instances completely different from trining and test set}
-% \begin{table}
-% \begin{tabular}{l l l}
-% \toprule
-% \textbf{Model} & \textbf{In-Distr Sensitivity} & \textbf{Out-Distr Sensitivity} \\
-% \midrule
-% Random Forest & 0.930 & 0.273 \\
-% Deep Class. + GNN encoding & 0.933 & 0.025 \\
-% Homogeneous NG encoding & 0.064 & 0.049 \\
-% Heterogeneous NG encoding & - & - \\
-% \bottomrule
-% \end{tabular}
-% \caption{Comparing sensitivity of the model with a selection of Solomon Instances.}
-% \end{table}
-\end{frame}
+%\begin{frame}{DNE-neighborhood vs learned neighborhood}
+%
+%
+%Picture neighborhood vs learned neighborhood - are the learned rather sparse or have more 1-entries then necessary?
+%%\red{Sensitivity means correctly guessed 1's right? Its impotant cause structure is sparse\\
+%%\red{In distribution is test set and out distribution is e.g. solomon instances, instances completely different from trining and test set}
+%% \begin{table}
+%% \begin{tabular}{l l l}
+%% \toprule
+%% \textbf{Model} & \textbf{In-Distr Sensitivity} & \textbf{Out-Distr Sensitivity} \\
+%% \midrule
+%% Random Forest & 0.930 & 0.273 \\
+%% Deep Class. + GNN encoding & 0.933 & 0.025 \\
+%% Homogeneous NG encoding & 0.064 & 0.049 \\
+%% Heterogeneous NG encoding & - & - \\
+%% \bottomrule
+%% \end{tabular}
+%% \caption{Comparing sensitivity of the model with a selection of Solomon Instances.}
+%% \end{table}
+%\end{frame}
\subsection{Discussion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -1174,7 +1202,7 @@ Picture neighborhood vs learned neighborhood - are the learned rather sparse or
\begin{frame}{Outlook}
\begin{itemize}\setlength\itemsep{1em}
\item Evaluate importance of input features
- \item Evaluate the learning success by using the learned neighborhood in a fully-fledged branch-price-and-cut algorithm.
+ \item Evaluate the learning success by using the learned neighborhood in a fully-fledged branch-price-and-cut algorithm
\item Extend the research on other VRP-variants
\item Use Learning in other parts of the branch-price-and-cut algorithm
\end{itemize}