IEEE Computational Intelligence Magazine - August 2022 - 34

Frontier
Research
Yansen Su, Zhongxiang Jin,
Ye Tian, and Xingyi Zhang
Anhui University, CHINA
Kay Chen Tan
The Hong Kong Polytechnic
University, HONG KONG SAR
Comparing the Performance of Evolutionary Algorithms for Sparse
Multi-Objective Optimization via a Comprehensive Indicator
Abstract
M
any real-world multi-objective
optimization problems (MOPs)
are characterized by a large
number of decision variables, where
the decision variables are mostly set to
zero in the Pareto optimal solutions.
Although some multi-objective evolutionary
algorithms (MOEAs) have been
tailored for large-scale MOPs in recent
years, most of them do not consider the
sparse nature of Pareto optimal solutions,
and their effectiveness to sparse MOPs
has not been investigated. Therefore, this
work aims to compare the performance
of MOEAs on sparse MOPs by suggesting
a comprehensive performance
indicator. In comparison to existing
indicators assessing the convergence and
diversity of a solution set according to
predefined reference points, the proposed
indicator can assess the convergence,
diversity, and sparsity without
using any reference point. Based on the
proposed indicator, an experiment is
conducted to compare the performance
of 11 state-of-the-art MOEAs on 60 test
instances taken from benchmark suites
and real-world applications. The statistical
results show that some MOEAs are
significantly better than the others for
solving sparse MOPs, and the proposed
indicator is effective for the performance
assessment on sparse MOPs.
I. Introduction
Multi-objective optimization problems
(MOPs) widely exist in scientific and
engineering areas, which contain two or
Digital Object Identifier 10.1109/MCI.2022.3180913
Date of current version: 19 July 2022
optimize each group of variables alternately
[2], [4]. While the variable grouping
strategies
for single-objective
more objectives to be optimized simultaneously
[1]. In the absence of a single
solution making all the objectives optimal,
there exist multiple Pareto optimal
solutions for an MOP, which make
diverse trade-offs between the conflicting
objectives. Since multiple optimal
solutions rather than a single one need
to be provided, solving MOPs is much
more difficult than solving single-objective
optimization problems. More seriously,
with the curse of dimensionality
in large-scale optimization, it is extremely
difficult for conventional MOEAs to
find multiple optimal solutions for largescale
MOPs [2].
To improve the efficiency of approximating
Pareto optimal solutions from
high-dimensional decision spaces, a
number of MOEAs have been tailored
for solving large-scale MOPs since 2013
[2]. By adopting the divide-and-conquer
strategy for large-scale single-objective
optimization [3], some early MOEAs for
large-scale MOPs divide the decision
variables randomly or heuristically and
Corresponding author: Ye Tian (e-mail: field910921@
gmail.com).
optimization do not preserve the population
diversity in the objective space,
some MOEAs group the decision variables
according to their contributions to
the convergence and diversity of the
population, showing high performance
on large-scale MOPs with complicated
landscapes [5], [6]. On the other hand,
some MOEAs aim to facilitate the solving
of large-scale MOPs by converting
the original problem into a small-scale
one, where quasi-optimal solutions can
be obtained with a few function evaluations
[7], [8]. Similarly, some MOEAs
adopt the dimensionality reduction
strategies used in machine learning to
directly reduce the decision space for an
improvement of the convergence speed
[9], [10]. In addition to the variable
grouping and dimensionality reduction
based MOEAs, some others suggest
novel variation operators [11], [12] or
probability models [13], [14] to search
for optimal solutions directly, which are
versatile for solving large-scale MOPs
with different variable linkages.
In spite of the promising performance
of existing large-scale MOEAs
on a variety of benchmark problems,
only few of them have been employed
to tackle the large-scale MOPs in realworld
applications. In addition to the
high budget of evaluations required by
these MOEAs, they do not
take the
characteristics of real-world applications
into account, such as the sparse nature of
Pareto optimal solutions. For the largescale
MOPs in many fields including
machine learning [15], data mining [16],
34 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2022
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