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Fix spacing issues in align environment

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Jip J. Dekker 2021-05-18 11:02:12 +10:00
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22
chapters/2_background.tex
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@ -700,8 +700,8 @@ using a set of linear equations over continuous variables. In general, a linear
program can be expressed in the form: program can be expressed in the form:
\begin{align*} \begin{align*}
\text{maximise } & \sum_{j=1}^{V} c_{j} x_{j} & \\ \text{maximise} \hspace{2em} & \sum_{j=1}^{V} c_{j} x_{j} & \\
\text{subject to } & l_{i} \leq \sum_{j=0}^{V} a_{ij} x_{j} \leq u_{i} & \forall_{i=1}^{C} \\ \text{subject to} \hspace{2em} & l_{i} \leq \sum_{j=0}^{V} a_{ij} x_{j} \leq u_{i} & \forall_{i=1}^{C} \\
& x_{i} \in \mathbb{R} & \forall_{i=1}^{V} & x_{i} \in \mathbb{R} & \forall_{i=1}^{V}
\end{align*} \end{align*}
@ -743,9 +743,11 @@ bound from an earlier solution, then we know that any discrete solutions
following from the linear program will be strictly worse than the incumbent. following from the linear program will be strictly worse than the incumbent.
Over the years \gls{lp} and \gls{mip} \glspl{solver} have developed immensely. Over the years \gls{lp} and \gls{mip} \glspl{solver} have developed immensely.
\Glspl{solver}, such as CBC \autocite{}, CPLEX \autocite{}, Gurobi \autocite{}, \Glspl{solver}, such as CBC \autocite{forrest-2020-cbc}, CPLEX
and SCIP \autocite{}, can solve many complex problems. It is therefore often \autocite{cplex-2020-cplex}, Gurobi \autocite{gurobi-2021-gurobi}, and SCIP
worthwhile to encode problem as an mixed integer program to find a solution. \autocite{gamrath-2020-scip}, can solve many complex problems. It is therefore
often worthwhile to encode problem as an mixed integer program to find a
solution.
\glspl{csp} can be often be encoded as mixed integer programs. This does, \glspl{csp} can be often be encoded as mixed integer programs. This does,
however, come with its challenges. Most \glspl{constraint} in a \minizinc\ model however, come with its challenges. Most \glspl{constraint} in a \minizinc\ model
@ -763,8 +765,8 @@ can then be rewritten as linear \glspl{constraint} using the \glspl{variable}
following model shows a integer program of this problem. following model shows a integer program of this problem.
\begin{align} \begin{align}
\text{maximise } & 0 & \\ \text{maximise} \hspace{2em} & 0 & \\
\text{subject to } & q_{i} \in \{1,\ldots{},n\} & \forall_{i=1}^{n} \\ \text{subject to} \hspace{2em} & q_{i} \in \{1,\ldots{},n\} & \forall_{i=1}^{n} \\
& y_{ij} \in \{0,1\} & \forall_{i=1}^{n} \forall_{j=1}^{n} \\ & y_{ij} \in \{0,1\} & \forall_{i=1}^{n} \forall_{j=1}^{n} \\
\label{line:back-mip-channel} & x_{i} = \sum_{j=1}^{n} j * y_{ij} & \forall_{i=1}^{n} \\ \label{line:back-mip-channel} & x_{i} = \sum_{j=1}^{n} j * y_{ij} & \forall_{i=1}^{n} \\
\label{line:back-mip-row} & \sum_{i=1}^{n} y_{ij} \leq 1 & \forall_{j=1}^{n} \label{line:back-mip-row} & \sum_{i=1}^{n} y_{ij} \leq 1 & \forall_{j=1}^{n}
@ -821,9 +823,9 @@ most efficient way to solve the problem.
\gls{sat} encoding for this problem is the following. \gls{sat} encoding for this problem is the following.
\begin{align} \begin{align}
\text{given } & n & \\ \text{given} \hspace{2em} & n & \\
\text{find } & q_{ij} \in \{\text{true}, \text{false}\} & \forall_{i=1}^{n}\forall_{j=1}^{n} \\ \text{find} \hspace{2em} & q_{ij} \in \{\text{true}, \text{false}\} & \forall_{i=1}^{n}\forall_{j=1}^{n} \\
\label{line:back-sat-at-least}\text{subject to } & \exists_{j=1}^{n} q_{ij} & \forall_{i=1}^{n} \\ \label{line:back-sat-at-least}\text{subject to} \hspace{2em} & \exists_{j=1}^{n} q_{ij} & \forall_{i=1}^{n} \\
\label{line:back-sat-row}& \neg q_{ij} \lor \neg q_{ik} & \forall_{i=1}^{n} \forall_{j=1}^{n} \forall_{k=j}^{n}\\ \label{line:back-sat-row}& \neg q_{ij} \lor \neg q_{ik} & \forall_{i=1}^{n} \forall_{j=1}^{n} \forall_{k=j}^{n}\\
\label{line:back-sat-col}& \neg q_{ij} \lor \neg q_{kj} & \forall_{i=1}^{n} \forall_{j=1}^{n} \forall_{k=i}^{n}\\ \label{line:back-sat-col}& \neg q_{ij} \lor \neg q_{kj} & \forall_{i=1}^{n} \forall_{j=1}^{n} \forall_{k=i}^{n}\\
\label{line:back-sat-diag1}& \neg q_{ij} \lor \neg q_{(i+k)(j+k)} & \forall_{i=1}^{n} \forall_{j=1}^{n} \forall_{k=1}^{min(n-i, n-j)}\\ \label{line:back-sat-diag1}& \neg q_{ij} \lor \neg q_{(i+k)(j+k)} & \forall_{i=1}^{n} \forall_{j=1}^{n} \forall_{k=1}^{min(n-i, n-j)}\\