Dekker1/dekker-phd-thesis
Archived
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Fix spacing issues in align environment

Browse Source
This commit is contained in:
Jip J. Dekker 2021-05-18 11:02:12 +10:00
parent cbec0bea7e
commit 1e64cf304d
No known key found for this signature in database
GPG Key ID: 517DF4A00618C9C3
1 changed files with 12 additions and 10 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

22
chapters/2_background.tex
View File

@ -700,8 +700,8 @@ using a set of linear equations over continuous variables. In general, a linear
program can be expressed in the form:
\begin{align*}
\text{maximise } & \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{maximise} \hspace{2em} & \sum_{j=1}^{V} c_{j} x_{j} & \\
\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}
\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.
Over the years \gls{lp} and \gls{mip} \glspl{solver} have developed immensely.
\Glspl{solver}, such as CBC \autocite{}, CPLEX \autocite{}, Gurobi \autocite{},
and SCIP \autocite{}, can solve many complex problems. It is therefore often
worthwhile to encode problem as an mixed integer program to find a solution.
\Glspl{solver}, such as CBC \autocite{forrest-2020-cbc}, CPLEX
\autocite{cplex-2020-cplex}, Gurobi \autocite{gurobi-2021-gurobi}, and SCIP
\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,
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.
\begin{align}
\text{maximise } & 0 & \\
\text{subject to } & q_{i} \in \{1,\ldots{},n\} & \forall_{i=1}^{n} \\
\text{maximise} \hspace{2em} & 0 & \\
\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} \\
\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}
@ -821,9 +823,9 @@ most efficient way to solve the problem.
\gls{sat} encoding for this problem is the following.
\begin{align}
\text{given } & n & \\
\text{find } & 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} \\
\text{given} \hspace{2em} & 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} \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-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)}\\