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Commit d8a81833 authored by Christian Tilk's avatar Christian Tilk
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......@@ -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}
......
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