begin fixes to paper

Author: hdavid16

paper/paper.tex

@@ -69,7 +69,7 @@ \subsection{Model}
 \end{align}
 \vskip 6pt
 
-\subsection{Linear GDP reformulation example}
+\subsection{Solution Technique: Reformulation to Mixed-Integer Program}
 The simplest example of a linear GDP system is given in \eqref{eq:ex} - \eqref{eq:y}, where $Y_i$ is a Boolean indicator variable that enforces the constraints in the disjunct ($Ax \le b$ or $Cx \le d$) when $true$.
 \vskip 6pt
 \begin{equation}
@@ -107,7 +107,7 @@ \subsection{Linear GDP reformulation example}
 \vskip 6pt
 
  \subsubsection{Big-M Reformulation}
- The Big-M reformulation for this problem is given by \eqref{eq:x}, \eqref{eq:ex_bigm1} - \eqref{eq:ex_bigm4}, where $M$ is a sufficiently large scalar that makes the particular constraint redundant when its indicator variable is not selected (i.e., $y_i = 0$). Note that the Boolean variables, $Y_i$, are replaced by binary variables, $y_i$. When the integrality constraint in Eq. \eqref{eq:ex_bigm4} is relaxed to $0 \leq x_1, x_2 \leq 1$, the resulting feasible region can be visualized by projecting the relaxed model onto the $x_1, x_2$ plane. This results in the region encapsulated by the dashed line in Figure \ref{fig:bigm}. It should be noted that the relaxed feasible region is not as tight as possible around the original feasible solution space. The choice of the large $M$ value determines the tightness of this relaxation, and the minimal value of $M$ for the optimal relaxation can be found through interval arithmetic when the model is linear. For nonlinear models, the tightest $M$ can be obtained by solving the maximization problem $\{\max h_{ik}(x): x \in X\}$. An alternate method for tight Big-M relaxations is given in \cite{TRESPALACIOS201598}.
+ The Big-M reformulation for this problem is given by \eqref{eq:ex_bigm1} - \eqref{eq:ex_bigm5}, where $M$ is a sufficiently large scalar that makes the particular constraint redundant when its indicator variable is not selected (i.e., $y_i = 0$). Note that the Boolean variables, $Y_i$, are replaced by binary variables, $y_i$. When the integrality constraint in Eq. \eqref{eq:ex_bigm4} is relaxed to $0 \leq x_1, x_2 \leq 1$, the resulting feasible region can be visualized by projecting the relaxed model onto the $x_1, x_2$ plane. This results in the region encapsulated by the dashed line in Figure \ref{fig:bigm}. It should be noted that the relaxed feasible region is not as tight as possible around the original feasible solution space. The choice of the large $M$ value determines the tightness of this relaxation, and the minimal value of $M$ for the optimal relaxation can be found through interval arithmetic when the model is linear. For nonlinear models, the tightest $M$ can be obtained by solving the maximization problem $\{\max h_{ik}(x): x \in X\}$. An alternate method for tight Big-M relaxations is given in \cite{TRESPALACIOS201598}.
 
 \begin{equation}
     \label{eq:ex_bigm1}
@@ -123,6 +123,10 @@ \subsection{Linear GDP reformulation example}
 \end{equation}
 \begin{equation}
     \label{eq:ex_bigm4}
+    0 \leq x \leq U
+\end{equation}
+\begin{equation}
+    \label{eq:ex_bigm5}
     y_1, y_2 \in \{0,1\}
 \end{equation}
 
@@ -134,24 +138,36 @@ \subsection{Linear GDP reformulation example}
 \end{figure}
 
 \subsubsection{Hull Reformulation}
-The Hull reformulation is given by \eqref{eq:x}, \eqref{eq:ex_bigm3} - \eqref{eq:ex_hull3}, which requires lifting the model to a higher-dimensional space. When projected to the original space, the continuous relaxation of the model is tighter than its Big-M equivalent \cite{grossmann_trespalacios_2013}. The reformulation relaxation can be visualized by the region encapsulated by the dashed line in Figure \ref{fig:chr}. Note that this reformulation provides a tighter relaxation than the Big-M reformulation in Figure \ref{fig:bigm}. Also note that describing the geometry of this relaxation is more complex than the Big-M relaxation, which is made possible by the increased number of constraints and variables in the model.
+The Hull reformulation is given by \eqref{eq:ex_hull0} - \eqref{eq:ex_hull6}, which requires lifting the model to a higher-dimensional space. When projected to the original space, the continuous relaxation of the model is tighter than its Big-M equivalent \cite{grossmann_trespalacios_2013}. The reformulation relaxation can be visualized by the region encapsulated by the dashed line in Figure \ref{fig:chr}. Note that this reformulation provides a tighter relaxation than the Big-M reformulation in Figure \ref{fig:bigm}. Also note that describing the geometry of this relaxation is more complex than the Big-M relaxation, which is made possible by the increased number of constraints and variables in the model.
 
 \begin{equation}
-    \label{eq:ex_hull1}
+    \label{eq:ex_hull0}
     Ax_1 \leq by_1
 \end{equation}
 \begin{equation}
-    \label{eq:ex_hull0}
+    \label{eq:ex_hull1}
     Cx_2 \leq dy_2
 \end{equation}
 \begin{equation}
     \label{eq:ex_hull2}
     x = x_1 + x_2
 \end{equation}
+\begin{equation}
+    \label{eq:ex_hull4}
+    y_1 + y_2 = 1
+\end{equation}
+\begin{equation}
+    \label{eq:ex_hull5}
+    0 \leq x \leq U
+\end{equation}
 \begin{equation}
     \label{eq:ex_hull3}
     0 \leq x_i \leq U y_i \quad \forall i \in \{1,2\}
 \end{equation}
+\begin{equation}
+    \label{eq:ex_hull6}
+    y_1, y_2 \in \{0,1\}
+\end{equation}
 
 \begin{figure}%[H]
     \centering
@@ -173,10 +189,10 @@ \subsubsection{Propositional Logic}
     (A \land B) \lor C & \text{ is replaced by } (A \lor C) \land (B \lor C)
 \end{align*}
 
-Once the logic propositions are converted to CNF, each clause can be converted into an algebraic constraint with the following equivalence (Note: any negated Boolean variables, $\neg Y_i$, are replaced with $1-y_i$ in the reformulation),
+Once the logic propositions are converted to CNF, each clause can be converted into an algebraic constraint with the following equivalence, where the set $I$ represents the subset boolean variables present in the clause, and the set $J$ represents the subset of boolean variables present in the clause in negated form,
 
 \begin{align*}
-    \bigvee_{i \in I} Y_i & \ \ \text{becomes} \ \ \sum_{i\in I} y_i \geq 1 \\
+    \bigvee_{i \in I} Y_i \bigvee_{j \in J} (\neg Y_j) & \ \ \text{becomes} \ \ \sum_{i \in I} y_i + \sum_{j \in J} (1-y_j) \geq 1 \\
 \end{align*}
 
 Alternate approaches exist for converting propositional logic statements into CNF, which involve preserving clause satisfiability rather than clause equivalence. These approaches prevent exponential size increase in clauses and yield logically consistent results \cite{jackson_sheridan_2005}.
@@ -288,16 +304,19 @@ \subsection{Example}
     \item Create the JuMP model and define the model variables and global constraints (mass balances).
 
 \begin{lstlisting}[language = Julia]
-using DisjunctiveProgramming, JuMP, HiGHS
+using DisjunctiveProgramming, HiGHS
 
 # create model
-m = JuMP.Model(HiGHS.Optimizer)
+m = GDPModel(HiGHS.Optimizer)
 # add variables to model
 @variable(m, 0 <= F[i = 1:7] <= 10)
 @variable(m, 0 <= CS <= CSmax)
 @variable(m, CRmin <= CR <= CRmax)
+# add logical variables to model
+@variable(m, YR[1:2], LogicalVariable)
+@variable(m, YS[1:2], LogicalVariable)
 
-# add constraints to model
+# add global constraints to model
 @constraints(m,
     begin
         F[1] == F[2] + F[3]
@@ -307,56 +326,32 @@ \subsection{Example}
 \end{lstlisting}
     \item Define the inner (nested) disjunction for the separation technologies in the superstructure using the \verb|@disjunction| macro.   
 \begin{lstlisting}[language = Julia]
-@disjunction(m,
-    begin
-        F[5] == β[:S1]*F[4]
-        CS == γ[:S1]
-    end,
-    begin
-        F[5] == β[:S2]*F[4]
-        CS == γ[:S2]
-    end,
-    reformulation = :big_m, # reformulation type
-    name = :YS # symbol for indicator variable
-)
-\end{lstlisting}
-    \item Define constraints in the outer disjunctions.
-\begin{lstlisting}[language = Julia]
-# define constraints in left disjunct
-R1_con = @constraints(m,
-    begin
-        F[6] == β[:R1]*F[2]
-        [i = 3:5], F[i] == 0
-        CR == γ[:R1]
-        CS == 0
-    end
-)
-
-# define constraints in right disjunct
-R2_con = @constraints(m,
-    begin
-        F[6] == β[:R2]*F[3]
-        CR == γ[:R2]      
-    end
-)
+# define constraints in left YS disjunct
+@constraint(m, F[5] == β[:S1]*F[4], DisjunctConstraint(YS[1]))
+@constraint(m, CS == γ[:S1], DisjunctConstraint(YS[1]))
+# define constraints in right YS disjunct
+@constraint(m, F[5] == β[:S2]*F[4], DisjunctConstraint(YS[2]))
+@constraint(m, CS == γ[:S2], DisjunctConstraint(YS[2]))
+# define disjunction (specify parent disjunct)
+@disjunction(m, YS, DisjunctConstraint(YR[2]))
 \end{lstlisting}
-    \item Build the main disjunction using the constraint blocks defined in (3) and the \verb|add_disjunction!| function. Note that the reformulated constraints for the nested disjunction are stored in the \verb|.ext| dictionary of the model under the name of the disjunction (\verb|:YS| in this case).
+    \item Define the outer disjunctions.
 \begin{lstlisting}[language = Julia]
-add_disjunction!(m,
-    R1_con,
-    (
-        R2_con, #general constraints in R2 disj.
-        m.ext[:YS] #reformulated inner disj.
-    ),
-    reformulation = :big_m, # reformulation type
-    name = :YR # symbol for indicator variable
-)
+# define constraints in left YR disjunct
+@constraint(m, F[6] == β[:R1]*F[2], DisjunctConstraint(YR[1]))
+@constraint(m, [i = 3:5], F[i] == 0, DisjunctConstraint(YR[1]))
+@constraint(m, CR == γ[:R1], DisjunctConstraint(YR[1]))
+@constraint(m, CS == 0, DisjunctConstraint(YR[1]))
+# define constraints in right YR disjunct
+@constraint(m, F[6] == β[:R2]*F[3], DisjunctConstraint(YR[2]))
+@constraint(m, CR == γ[:R2], DisjunctConstraint(YR[2]))  
+# define disjunction
+@disjunction(m, YR)    
 \end{lstlisting}
     \item Add the selection logical constraints using the \verb|choose!| function. The first constraint enforces that only one reactor is selected (i.e., $Y_{R_1} \ \underline{\vee} \ Y_{R_2}$). The second constraint enforces that the separation system be defined only if the second reactor ($R_2$) is selected. This constraint is equivalent to the proposition $Y_{R_2} \Leftrightarrow Y_{S_1} \ \underline{\vee} \ Y_{S_2}$.
 \begin{lstlisting}[language = Julia]
-YR, YS = m[:YR], m[:YS]
-choose!(m, 1, YR[1], YR[2]; mode = :exactly)
-choose!(m, YR[2], YS[1], YS[2]; mode = :exactly)
+@constraint(m, YR in Exactly(1))
+@constraint(m, YS in Exactly(YR[2]))
 \end{lstlisting}
     \item Add the objective function and optimize. 
 \begin{lstlisting}[language = Julia]

paper/ref.bib

@@ -198,7 +198,7 @@ @article{jackson_sheridan_2005
 } 
 
 @article{dunning_huchette_lubin_2017,
-title={Jump: A modeling language for mathematical optimization},
+title={JuMP: A modeling language for mathematical optimization},
 volume={59},
 number={2},
 journal={SIAM Review},