2,626 research outputs found
Multiplicative Approximations, Optimal Hypervolume Distributions, and the Choice of the Reference Point
Many optimization problems arising in applications have to consider several
objective functions at the same time. Evolutionary algorithms seem to be a very
natural choice for dealing with multi-objective problems as the population of
such an algorithm can be used to represent the trade-offs with respect to the
given objective functions. In this paper, we contribute to the theoretical
understanding of evolutionary algorithms for multi-objective problems. We
consider indicator-based algorithms whose goal is to maximize the hypervolume
for a given problem by distributing {\mu} points on the Pareto front. To gain
new theoretical insights into the behavior of hypervolume-based algorithms we
compare their optimization goal to the goal of achieving an optimal
multiplicative approximation ratio. Our studies are carried out for different
Pareto front shapes of bi-objective problems. For the class of linear fronts
and a class of convex fronts, we prove that maximizing the hypervolume gives
the best possible approximation ratio when assuming that the extreme points
have to be included in both distributions of the points on the Pareto front.
Furthermore, we investigate the choice of the reference point on the
approximation behavior of hypervolume-based approaches and examine Pareto
fronts of different shapes by numerical calculations
Diversity comparison of Pareto front approximations in many-objective optimization
Diversity assessment of Pareto front approximations is an important issue in the stochastic multiobjective optimization community. Most of the diversity indicators in the literature were designed to work for any number of objectives of Pareto front approximations in principle, but in practice many of these indicators are infeasible or not workable when the number of objectives is large. In this paper, we propose a diversity comparison indicator (DCI) to assess the diversity of Pareto front approximations in many-objective optimization. DCI evaluates relative quality of different Pareto front approximations rather than provides an absolute measure of distribution for a single approximation. In DCI, all the concerned approximations are put into a grid environment so that there are some hyperboxes containing one or more solutions. The proposed indicator only considers the contribution of different approximations to nonempty hyperboxes. Therefore, the computational cost does not increase exponentially with the number of objectives. In fact, the implementation of DCI is of quadratic time complexity, which is fully independent of the number of divisions used in grid. Systematic experiments are conducted using three groups of artificial Pareto front approximations and seven groups of real Pareto front approximations with different numbers of objectives to verify the effectiveness of DCI. Moreover, a comparison with two diversity indicators used widely in many-objective optimization is made analytically and empirically. Finally, a parametric investigation reveals interesting insights of the division number in grid and also offers some suggested settings to the users with different preferences
Biobjective Performance Assessment with the COCO Platform
This document details the rationales behind assessing the performance of
numerical black-box optimizers on multi-objective problems within the COCO
platform and in particular on the biobjective test suite bbob-biobj. The
evaluation is based on a hypervolume of all non-dominated solutions in the
archive of candidate solutions and measures the runtime until the hypervolume
value succeeds prescribed target values
An Analysis on Selection for High-Resolution Approximations in Many-Objective Optimization
This work studies the behavior of three elitist multi- and many-objective
evolutionary algorithms generating a high-resolution approximation of the
Pareto optimal set. Several search-assessment indicators are defined to trace
the dynamics of survival selection and measure the ability to simultaneously
keep optimal solutions and discover new ones under different population sizes,
set as a fraction of the size of the Pareto optimal set.Comment: apperas in Parallel Problem Solving from Nature - PPSN XIII,
Ljubljana : Slovenia (2014
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