4 research outputs found
Constructing Low Star Discrepancy Point Sets with Genetic Algorithms
Geometric discrepancies are standard measures to quantify the irregularity of
distributions. They are an important notion in numerical integration. One of
the most important discrepancy notions is the so-called \emph{star
discrepancy}. Roughly speaking, a point set of low star discrepancy value
allows for a small approximation error in quasi-Monte Carlo integration. It is
thus the most studied discrepancy notion.
In this work we present a new algorithm to compute point sets of low star
discrepancy. The two components of the algorithm (for the optimization and the
evaluation, respectively) are based on evolutionary principles. Our algorithm
clearly outperforms existing approaches. To the best of our knowledge, it is
also the first algorithm which can be adapted easily to optimize inverse star
discrepancies.Comment: Extended abstract appeared at GECCO 2013. v2: corrected 3 numbers in
table
Discrepancy-based Evolutionary Diversity Optimization
Diversity plays a crucial role in evolutionary computation. While diversity
has been mainly used to prevent the population of an evolutionary algorithm
from premature convergence, the use of evolutionary algorithms to obtain a
diverse set of solutions has gained increasing attention in recent years.
Diversity optimization in terms of features on the underlying problem allows to
obtain a better understanding of possible solutions to the problem at hand and
can be used for algorithm selection when dealing with combinatorial
optimization problems such as the Traveling Salesperson Problem. We explore the
use of the star-discrepancy measure to guide the diversity optimization process
of an evolutionary algorithm.
In our experimental investigations, we consider our discrepancy-based
diversity optimization approaches for evolving diverse sets of images as well
as instances of the Traveling Salesperson problem where a local search is not
able to find near optimal solutions. Our experimental investigations comparing
three diversity optimization approaches show that a discrepancy-based diversity
optimization approach using a tie-breaking rule based on weighted differences
to surrounding feature points provides the best results in terms of the star
discrepancy measure
Heuristic Approaches to Obtain Low-Discrepancy Point Sets via Subset Selection
Building upon the exact methods presented in our earlier work [J. Complexity,
2022], we introduce a heuristic approach for the star discrepancy subset
selection problem. The heuristic gradually improves the current-best subset by
replacing one of its elements at a time. While we prove that the heuristic does
not necessarily return an optimal solution, we obtain very promising results
for all tested dimensions. For example, for moderate point set sizes in dimension 6, we obtain point sets with star
discrepancy up to 35% better than that of the first points of the Sobol'
sequence. Our heuristic works in all dimensions, the main limitation being the
precision of the discrepancy calculation algorithms.
We also provide a comparison with a recent energy functional introduced by
Steinerberger [J. Complexity, 2019], showing that our heuristic performs better
on all tested instances