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diff --git a/SimpleDocument.pdf b/SimpleDocument.pdf
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diff --git a/SimpleDocument.synctex.gz b/SimpleDocument.synctex.gz
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diff --git a/SimpleDocument.tex b/SimpleDocument.tex
index b41aba8..eecbaa9 100644
--- a/SimpleDocument.tex
+++ b/SimpleDocument.tex
@@ -19,7 +19,7 @@
\title{Gran Compendio OLI}
\author{\href{https://github.com/meschio94/Gran-Compendio-OLI}{Meschio}}
-\date{2021\\V1.0a}
+\date{2021\\V1.0b}
\begin{document}
\maketitle
@@ -408,9 +408,21 @@ Procediamo infine ad avere gli 0 nelle rispettive colonne della nostra base, in
\subsubsection{Duale del problema}
Come trasformiamo il nostro sistema in un problema duale ?\\
-Per prima cosa dobbiamo scegliere una direzione per i segni, tutti devono andare nella stessa direzione (non va considerata la riga finale delle $x_i \geqq 0$ ), Esempio:\\
+Per prima cosa dobbiamo scegliere una direzione per i segni, tutti devono andare nella stessa direzione (non va considerata la riga finale delle $x_i \geqq 0$ ), possiamo seguire questa tabella per indicazioni più rigorose:\\
+\begin{center}
+\raisebox{-.5\height}{\includegraphics[width=8cm]{immagini/DualeProblema0.jpg}}
+\raisebox{-.5\height}{\includegraphics[width=8cm]{immagini/DualeProblema0a.jpg}}
+\end{center}
+\begin{itemize}
+\item se i vincoli del primale sono: \textbf{$a_i^Tx \geqq b_i$},nel duale arriveremo dal basso della regione ammissibile, le slack di conseguenza saranno \textbf{non negative}. Nel duale staremo massimizzando, percui le nostre corrispettive variabili (moltiplicative) dovranno essere \textbf{non negative}.
+\item se i vincoli del primale sono \textbf{$a_i^Tx \leqq b_i$},le slack dei vincoli dovranno essere \textbf{non positive}. Nel duale staremo massimizzando, percui le nostre corrispettive variabili (moltiplicative) dovranno essere \textbf{non positive}.
+\item se i vincoli del primale sono \textbf{$a_i^Tx = b_i$}, le slack dei vincoli potranno avere qualsiasi valore. Le corrispondenti variabili (moltiplicative) potranno avere qualsiasi valore.
+\end{itemize}
+
+Esempio:\\
+
\begin{align*}
min: &2x_1 + 3x_2 + x_3 & & min:& &2x_1 + 3x_2 + x_3\\
&-x_1 + 3x_2 - 2x_3 \geqq 8 & \Rightarrow & & &-x_1 + 3x_2 - 2x_3 \geqq 8\\
@@ -418,6 +430,11 @@ min: &2x_1 + 3x_2 + x_3 & & min:& &2x_1 + 3x_2 + x_3\
&x_1, x_2,x_3 \geqq 0 & & & &x_1, x_2,x_3 \geqq 0\\
\end{align*}
+Possiamo sempre ricondurci alla forma canonica o standard prima di scrivere il duale:
+\begin{center}
+\raisebox{-.5\height}{\includegraphics[width=8cm]{immagini/DualeProblema0b.jpg}}
+\end{center}
+
Ordiniamo il sistema per una maggiore chiarezza:\\
\begin{center}
@@ -1399,7 +1416,7 @@ Soluzione:\\
Given a PLC problem with $n$ variables and $m$ constraints, a basic matrix is: (a) a collection of $m$ constraints; (b) none of the above answers.; (c) a square $m$x$m$ matrix with value 1 on the main diagonal; (d) a collection of $n-m$ columns of the constraint matrix; (e) a square submatrix of the constraint matrix, that can be inverted.
}%
}%
-\\
+\\/
Soluzione:\\
(B)
\end{document}
diff --git a/SimpleDocument.toc b/SimpleDocument.toc
index 5d328da..263bb1c 100644
--- a/SimpleDocument.toc
+++ b/SimpleDocument.toc
@@ -19,46 +19,46 @@
\contentsline {subsubsection}{\numberline {3.2.2}Simplesso Duale}{16}{subsubsection.3.2.2}%
\contentsline {subsubsection}{\numberline {3.2.3}2Fasi}{17}{subsubsection.3.2.3}%
\contentsline {subsubsection}{\numberline {3.2.4}Duale del problema}{19}{subsubsection.3.2.4}%
-\contentsline {subsection}{\numberline {3.3}Rappresentazione Grafica}{20}{subsection.3.3}%
+\contentsline {subsection}{\numberline {3.3}Rappresentazione Grafica}{21}{subsection.3.3}%
\contentsline {subsection}{\numberline {3.4}ILP}{22}{subsection.3.4}%
\contentsline {subsubsection}{\numberline {3.4.1}Tagli di Gomory}{22}{subsubsection.3.4.1}%
-\contentsline {subsection}{\numberline {3.5}Esercizi Particolari}{24}{subsection.3.5}%
-\contentsline {subsubsection}{\numberline {3.5.1}Simplesso con variabili free}{24}{subsubsection.3.5.1}%
-\contentsline {subsubsection}{\numberline {3.5.2}Problema PLC con simplesso e gomory in salsa teriyaki}{26}{subsubsection.3.5.2}%
-\contentsline {subsubsection}{\numberline {3.5.3}Senstivity Analysis}{28}{subsubsection.3.5.3}%
-\contentsline {section}{\numberline {4}Grafi}{30}{section.4}%
-\contentsline {subsection}{\numberline {4.1}GT}{30}{subsection.4.1}%
-\contentsline {subsubsection}{\numberline {4.1.1}Dijkstra}{30}{subsubsection.4.1.1}%
-\contentsline {subsubsection}{\numberline {4.1.2}Shortest Path Tree}{32}{subsubsection.4.1.2}%
-\contentsline {subsection}{\numberline {4.2}SST}{33}{subsection.4.2}%
-\contentsline {subsubsection}{\numberline {4.2.1}Cenni di Base}{33}{subsubsection.4.2.1}%
-\contentsline {subsubsection}{\numberline {4.2.2}SST Prim's}{34}{subsubsection.4.2.2}%
-\contentsline {subsubsection}{\numberline {4.2.3}SST Da Matrice trovare la soluzione ottimale}{35}{subsubsection.4.2.3}%
-\contentsline {subsection}{\numberline {4.3}Max Flow}{36}{subsection.4.3}%
-\contentsline {subsubsection}{\numberline {4.3.1}Cenni di Base}{36}{subsubsection.4.3.1}%
-\contentsline {subsubsection}{\numberline {4.3.2}Flow Network}{37}{subsubsection.4.3.2}%
-\contentsline {subsection}{\numberline {4.4}DP}{39}{subsection.4.4}%
-\contentsline {subsubsection}{\numberline {4.4.1}DP Knapsack 0-1 Dynamic Programming}{39}{subsubsection.4.4.1}%
-\contentsline {subsubsection}{\numberline {4.4.2}DP: SSP Bellman's-Ford}{41}{subsubsection.4.4.2}%
-\contentsline {subsection}{\numberline {4.5}ILP}{43}{subsection.4.5}%
-\contentsline {subsubsection}{\numberline {4.5.1}ILP standard B\&B}{43}{subsubsection.4.5.1}%
-\contentsline {subsubsection}{\numberline {4.5.2}ILP esercizio standard B\&B}{47}{subsubsection.4.5.2}%
-\contentsline {subsubsection}{\numberline {4.5.3}ILP esercizio B\&B 0-1}{49}{subsubsection.4.5.3}%
-\contentsline {section}{\numberline {5}GPLK}{51}{section.5}%
-\contentsline {subsection}{\numberline {5.1}Dal modello al codice}{51}{subsection.5.1}%
-\contentsline {subsubsection}{\numberline {5.1.1}Modello Easy}{51}{subsubsection.5.1.1}%
-\contentsline {subsubsection}{\numberline {5.1.2}Caso Particolare 1 Graffa}{52}{subsubsection.5.1.2}%
-\contentsline {subsubsection}{\numberline {5.1.3}Caso Particolare 2 Sommatoria doppio insieme}{54}{subsubsection.5.1.3}%
-\contentsline {subsection}{\numberline {5.2}Dal codice al modello}{55}{subsection.5.2}%
-\contentsline {subsubsection}{\numberline {5.2.1}Normale}{55}{subsubsection.5.2.1}%
-\contentsline {section}{\numberline {6}Domande varie di teoria}{56}{section.6}%
-\contentsline {subsection}{\numberline {6.1}PLC degenerate}{56}{subsection.6.1}%
-\contentsline {subsection}{\numberline {6.2}PLC sensitivty}{57}{subsection.6.2}%
-\contentsline {subsection}{\numberline {6.3}B\&B}{57}{subsection.6.3}%
-\contentsline {subsection}{\numberline {6.4}PLC minimization}{57}{subsection.6.4}%
-\contentsline {subsection}{\numberline {6.5}cutting Plane}{57}{subsection.6.5}%
-\contentsline {subsection}{\numberline {6.6}PLC dual}{58}{subsection.6.6}%
-\contentsline {subsection}{\numberline {6.7}Dijkstra}{58}{subsection.6.7}%
-\contentsline {subsection}{\numberline {6.8}B\&B knapsack}{58}{subsection.6.8}%
-\contentsline {subsection}{\numberline {6.9}Branch \& cut}{58}{subsection.6.9}%
-\contentsline {subsection}{\numberline {6.10}PLC}{59}{subsection.6.10}%
+\contentsline {subsection}{\numberline {3.5}Esercizi Particolari}{25}{subsection.3.5}%
+\contentsline {subsubsection}{\numberline {3.5.1}Simplesso con variabili free}{25}{subsubsection.3.5.1}%
+\contentsline {subsubsection}{\numberline {3.5.2}Problema PLC con simplesso e gomory in salsa teriyaki}{27}{subsubsection.3.5.2}%
+\contentsline {subsubsection}{\numberline {3.5.3}Senstivity Analysis}{29}{subsubsection.3.5.3}%
+\contentsline {section}{\numberline {4}Grafi}{31}{section.4}%
+\contentsline {subsection}{\numberline {4.1}GT}{31}{subsection.4.1}%
+\contentsline {subsubsection}{\numberline {4.1.1}Dijkstra}{31}{subsubsection.4.1.1}%
+\contentsline {subsubsection}{\numberline {4.1.2}Shortest Path Tree}{33}{subsubsection.4.1.2}%
+\contentsline {subsection}{\numberline {4.2}SST}{34}{subsection.4.2}%
+\contentsline {subsubsection}{\numberline {4.2.1}Cenni di Base}{34}{subsubsection.4.2.1}%
+\contentsline {subsubsection}{\numberline {4.2.2}SST Prim's}{35}{subsubsection.4.2.2}%
+\contentsline {subsubsection}{\numberline {4.2.3}SST Da Matrice trovare la soluzione ottimale}{36}{subsubsection.4.2.3}%
+\contentsline {subsection}{\numberline {4.3}Max Flow}{37}{subsection.4.3}%
+\contentsline {subsubsection}{\numberline {4.3.1}Cenni di Base}{37}{subsubsection.4.3.1}%
+\contentsline {subsubsection}{\numberline {4.3.2}Flow Network}{38}{subsubsection.4.3.2}%
+\contentsline {subsection}{\numberline {4.4}DP}{40}{subsection.4.4}%
+\contentsline {subsubsection}{\numberline {4.4.1}DP Knapsack 0-1 Dynamic Programming}{40}{subsubsection.4.4.1}%
+\contentsline {subsubsection}{\numberline {4.4.2}DP: SSP Bellman's-Ford}{42}{subsubsection.4.4.2}%
+\contentsline {subsection}{\numberline {4.5}ILP}{44}{subsection.4.5}%
+\contentsline {subsubsection}{\numberline {4.5.1}ILP standard B\&B}{44}{subsubsection.4.5.1}%
+\contentsline {subsubsection}{\numberline {4.5.2}ILP esercizio standard B\&B}{48}{subsubsection.4.5.2}%
+\contentsline {subsubsection}{\numberline {4.5.3}ILP esercizio B\&B 0-1}{50}{subsubsection.4.5.3}%
+\contentsline {section}{\numberline {5}GPLK}{52}{section.5}%
+\contentsline {subsection}{\numberline {5.1}Dal modello al codice}{52}{subsection.5.1}%
+\contentsline {subsubsection}{\numberline {5.1.1}Modello Easy}{52}{subsubsection.5.1.1}%
+\contentsline {subsubsection}{\numberline {5.1.2}Caso Particolare 1 Graffa}{53}{subsubsection.5.1.2}%
+\contentsline {subsubsection}{\numberline {5.1.3}Caso Particolare 2 Sommatoria doppio insieme}{55}{subsubsection.5.1.3}%
+\contentsline {subsection}{\numberline {5.2}Dal codice al modello}{56}{subsection.5.2}%
+\contentsline {subsubsection}{\numberline {5.2.1}Normale}{56}{subsubsection.5.2.1}%
+\contentsline {section}{\numberline {6}Domande varie di teoria}{57}{section.6}%
+\contentsline {subsection}{\numberline {6.1}PLC degenerate}{57}{subsection.6.1}%
+\contentsline {subsection}{\numberline {6.2}PLC sensitivty}{58}{subsection.6.2}%
+\contentsline {subsection}{\numberline {6.3}B\&B}{58}{subsection.6.3}%
+\contentsline {subsection}{\numberline {6.4}PLC minimization}{58}{subsection.6.4}%
+\contentsline {subsection}{\numberline {6.5}cutting Plane}{58}{subsection.6.5}%
+\contentsline {subsection}{\numberline {6.6}PLC dual}{59}{subsection.6.6}%
+\contentsline {subsection}{\numberline {6.7}Dijkstra}{59}{subsection.6.7}%
+\contentsline {subsection}{\numberline {6.8}B\&B knapsack}{59}{subsection.6.8}%
+\contentsline {subsection}{\numberline {6.9}Branch \& cut}{59}{subsection.6.9}%
+\contentsline {subsection}{\numberline {6.10}PLC}{60}{subsection.6.10}%
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