start documenting optimiser

doc/user_guide/introduction.tex

@@ -66,7 +66,8 @@ The following example is exemplifying this fact:
 \begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}lrrrr}
 %\toprule
 \hline
-      \tabhead{Distribution & Particles  & Mesh & Greens Function & Time steps}
+%      \tabhead{Distribution & Particles  & Mesh & Greens Function & Time steps}
+       Distribution & Particles  & Mesh & Greens Function & Time steps
 %      \midrule
 \hline
       Gauss 3D & $10^8$ & $1024^3$ & Integrated  & 10 \\

doc/user_guide/optimiser.tex

@@ -4,5 +4,137 @@
 \label{chp:moo}
 \index{Multi Objective Optimisation|(}
 
+Optimization methods deal with finding a feasible set of solutions
+  corresponding to extreme values of some specific criteria.
+  Problems consisting of more than one criterion are called
+  \textit{multi-objective optimization problems}.
+Multiple objectives arise naturally in many real world optimization problems,
+  such as portfolio optimization, design, planning and many more
+  \cite{pgnl:06,zepv:00,gala:98,yrss:09,basi:05}.
+It is important to stress that multi-objective problems are in general harder
+  and more expensive to solve than single-objective optimization problems.
+
+In this chapter we introduce multi-objective optimization problems and discuss
+  techniques for their solution with an emphasis on evolutionary algorithms.
+
+
+\section{Definition}
+
+As with single-objective optimization problems, multi-object optimization
+  problems consist of a solution vector and optionally a number of equality
+  and inequality constraints.
+Formally, a general multi-objective optimization problem has the form
+%
+\begin{align}
+  \text{ min} \quad & \quad f_m({\bf x}), ~& m &= \{1, \dots, M\} \label{eq:moop:obj}\\
+  \text{s.t.} \quad & \quad g_j({\bf x}) \geq 0, & j &= \{1, \dots, J\}
+  \label{eq:moop:constr}\\
+  \quad & \quad  -\infty \leq x_i^L \leq {\bf x}=x_i \leq x_i^U \leq \infty,& i &= \{0, \dots, n \}
+  \label{eq:moop:dvar} \text{.}
+\end{align}
+%
+The $M$ objectives (\ref{eq:moop:obj}) are minimized, subject to $J$
+  inequality constraints (\ref{eq:moop:constr}).
+An $n$-vector (\ref{eq:moop:dvar}) contains all the design variables with
+  appropriate lower and upper bounds, constraining the design space.
+
+In contrast to single-objective optimization the objective functions span
+  a multi-dimensional space in addition to the design variable space --
+  for each point in design space there exists a point in objective space.
+The mapping from the $n$ dimensional design space to the $M$ dimensional
+  objective space visualized in Figure~\ref{fig:des_to_obj} is often
+  non-linear.
+This impedes the search for optimal solutions and increases the computational
+  cost as a result of expensive objective function evaluation.
+Additionally, depending in which of the two spaces the algorithm uses to
+  determine the next step, it can be difficult to assure an even sampling of
+  both spaces simultaneously.
+%
+\begin{figure}
+  \begin{center}
+    \begin{tikzpicture}
+%      \input{../figures/tex/design_objective_space}
+    \end{tikzpicture}
+  \end{center}
+  \caption{The (often non-linear) mapping $f : \mathbb{R}^n \rightarrow
+    \mathbb{R}^M$ from design to objective space. The dashed lines represent
+    the constraints in design space.
+    %and the set of solutions (Pareto front) in objective space.
+    }
+  \label{fig:des_to_obj}
+\end{figure}
+\nomenclature{$\mathbb{R}$}{reel numbers}
+
+A special subset of multi-objective optimization problems where all objectives
+  and constraints are linear, called \textit{Multi-objective linear programs},
+  exhibit formidable theoretical properties that facilitate convergence proofs.
+In this thesis we strive to address arbitrary multi-objective optimization
+  problems with non-linear constraints and objectives.
+No general convergence proofs are readily available for these cases.
+
+
+\section{Pareto Optimality}
+
+In most multi-objective optimization problems we have to deal with conflicting
+  objectives.
+Two objectives are conflicting if they possess different minima.
+If all the mimima of all objectives coincide the multi-objective optimization
+  problem has only one solution.
+To facilitate comparing solutions we define a partial ordering relation on
+  candidate solutions based on the concept of dominance.
+A solution is said to dominate another solution if it is no worse than the
+  other solution in all objectives and if it is strictly better in at least
+  one objective.
+A more formal description of the dominance relation is given in
+  Definition~\ref{def:dom}~\cite{deb:09}.
+
+\begin{mydef} \label{def:dom}
+A point $\mathbf{x}_1$ is dominating $\mathbf{x}_2$, if both properties
+\begin{itemize}
+  \item $f_m(\mathbf{x}_1) \geq f_m(\mathbf{x}_2) \text{,} \;\; \forall m \in
+    \{ 1, \dots, M \}$
+  \item $f_m(\mathbf{x}_1) > f_m(\mathbf{x}_2) \text{,} \;\; \exists m \in
+    \{1, \dots, M\}$
+\end{itemize}
+hold. We denote this as $\mathbf{x}_1 \preceq \mathbf{x}_2$.
+\end{mydef}
+\nomenclature{$\preceq$}{dominance relation operator}
+
+The properties of the dominance relation include transitivity
+%
+\begin{equation*}
+  x_1 \preceq x_2 \wedge x_2 \preceq x_3 \Rightarrow x_1 \preceq x_3 \text{,}
+\end{equation*}
+%
+  and asymmetricity, which is necessary for an unambiguous order relation
+%
+\begin{equation*}
+  x_1 \preceq x_2 \Rightarrow x_2 \not\preceq x_1 \text{.}
+\end{equation*}
+%
+
+Using the concept of dominance, the sought-after set of Pareto optimal
+  solution points can be approximated iteratively as the set of non-dominated
+  solutions.
+
+The problem of deciding if a point truly belongs to the Pareto set is NP-hard.
+As shown in Figure~\ref{fig:pareto-def} there exist ``weaker'' formulations of
+  Pareto optimality.
+Of special interest is the result shown in \cite{paya:01}, where the authors
+  present a polynomial (in the input size of the problem and $1/\varepsilon$)
+  algorithm for finding an approximation, with accuracy $\varepsilon$, of the
+  Pareto set for database queries.
+
+  \begin{figure}[tp]
+  \begin{center}
+%    \includegraphics[angle=270,width=0.85\linewidth]{opt/pareto-defs.pdf}
+  \end{center}
+  \caption{Various definitions regarding Pareto optimality.}
+  \label{fig:pareto-def}
+\end{figure}
+
+
+
+
 
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