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LaTeX/thesis.bib
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year = {2016},
url = {https://arxiv.org/abs/1607.07892}
}
@inproceedings{englander2014tuning,
title={Tuning monotonic basin hopping: improving the efficiency of stochastic search as applied to low-thrust trajectory optimization},
author={Englander, Jacob A and Englander, Arnold C},
booktitle={International Symposium on Space Flight Dynamics 2014},
number={GSFC-E-DAA-TN14154},
year={2014}
}

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LaTeX/thesis.tex
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\label{mbh_flow}
\end{figure}
\subsection{Random Mission Generation}
\subsection{Random Mission Generation}\label{random_gen_section}
At a basic level, the algorithm needs to produce a mission guess (represented by all
of the values described in Section~\ref{inner_loop_section}) that contains random
@@ -872,9 +872,87 @@
matrix of uniform random numbers within that bound.
\subsection{Monotonic Basin Hopping}\label{mbh_subsection}
Outline the MBH algorithm, going into detail at each step. Mention the long-tailed PDF being
used and go into quite a bit of detail. Englander's paper on the MBH algorithm specifically
should be a good guide. Mention validation.
Now that a generator has been developed for mission guesses, we can build the
monotonic basin hopping algorithm. Since the implementation of the MBH algorithm
used in this paper differs from the standard implementation, the standard version
won't be described here. Instead, the variation used in this paper, with some
performance improvements, will be considered.
The aim of a monotonic basin hopping algorithm is to provide an efficient method for
completely traversing a large search space and providing many seed values within the
space for an ''inner loop`` solver or optimizer. These solutions are then perturbed
slightly, in order to provide higher fidelity searching in the space near valid
solutions in order to fully explore the vicinity of discovered local minima. This
makes it an excellent algorithm for problems with a large search space, including
several clusters of local minima, such as this application.
The algorithm contains two loops, the size of each of which can be independently
modified (generally by specifying a ''patience value``, or number of loops to
perform, for each) to account for trade-offs between accuracy and performance depending on
mission needs and the unique qualities of a certain search space.
The first loop, the ''search loop``, first calls the random mission generator. This
generator produces two random missions as described in
Section~\ref{random_gen_section} that differ only in that one contains random flyby
velocities and control thrusts and the other contains Lambert's-solved flyby
velocities and zero control thrusts. For each of these guesses, the NLP solver is
called. If either of these mission guesses have converged onto a valid solution, the
lower loop, the ''drill loop`` is entered for the valid solution. After the
convergence checks and potentially drill loops are performed, if a valid solution
has been found, this solution is stored in an archive. If the solution found is
better than the current best solution in the archive (as determined by a
user-provided cost function of fuel usage, $C_3$ at launch, and $v-\infty$ at
arrival) then the new solution replaces the current best solution and the loop is
repeated. Taken by itself, the search loop should quickly generate enough random
mission guesses to find all ''basins`` or areas in the solution space with valid
trajectories, but never attempts to more thoroughly explore the space around valid
solutions within these basins.
The drill loop, then, is used for this purpose. For the first step of the drill
loop, the current solution is saved as the ''basin solution``. If it's better than
the current best, it also replaces the current best solution. Then, until the
stopping condition has been met (generally when the ''drill counter`` has reached
the ''drill patience`` value) the current solution is perturbed slightly by adding
or subtracting a small random value to the components of the mission.
The performance of this perturbation in terms of more quickly converging upon the
true minimum of that particular basin, as described in detail by
Englander\cite{englander2014tuning}, is highly dependent on the distribution
function used for producing these random perturbations. While the intuitive choice
of a simple Gaussian distribution would make sense to use, it has been found that a
long-tailed distribution, such as a Cauchy distribution or a Pareto distribution is
more robust in terms of well chose boundary conditions and initial seed solutions as
well as more performant in time required to converge upon the minimum for that basin.
Because of this, the perturbation used in this implementation follows a
bi-directional, long-tailed Pareto distribution generated by the following
probability density function:
\begin{equation}
1 +
\left[ \frac{s}{\epsilon} \right] \cdot
\left[ \frac{\alpha - 1}{\frac{\epsilon}{\epsilon + r}^{-\alpha}} \right]
\end{equation}
Where $s$ is a random array of signs (either plus one or minus one) with dimension
equal to the perturbed variable and bounds of -1 and 1, $r$ is a uniformly
distributed random array with dimension equal to the perturbed variable and bounds
of 0 and 1, $\epsilon$ is a small value (nominally set to $1e-10$), and $\alpha$ is
a tuning parameter to determine the size of the tails and width of the distribution
set to $1.01$, but easily tunable.
The perturbation function, then steps through each parameter of the mission,
generating a new mission guess with the parameters modified by the above Pareto
distribution. After this perturbation, the NLP solver is then called again to find
a valid solution in the vicinity of this new guess. If the solution is better than
the current basin solution, it replaces that value and the drill counter is reset to
zero. If it is better than the current total best, it replaces that value as well.
Otherwise, the drill counter increments and the process is repeated. Therefore, the
drill patience allows the mission designer to determine a maximum number of
iterations to perform without any improvements in a row before ending a given drill
loop. This process can be repeated essentially ''search patience`` number of times
in order to fully traverse all basins.
\chapter{Results Analysis} \label{results}
Simply highlight that the algorithm was tested on a sample trajectory to Saturn.