IEEE Computational Intelligence Magazine - November 2019 - 53

them in the hope of mixing their good
features to produce new and hopefully
better offspring solutions. Occasional
mutations of offspring solutions ensure
that the evolving population does not
easily get stuck prematurely to sub-optimal regions and also help achieve locally
better solutions. Starting with Holland's
O (N 3) (where N is the population size)
schema processing argument in 1975 [1]
to Goldberg's implicit parallelism argument [2] in 1989 to more theoretical
studies in the EA field have all provided
reasons for the working of EAs.
Early EA researchers interested in solving multi-objective optimization problems
[2]-[6] have realized that in addition to
converging towards a single Pareto-optimal solution, a diversity-preserving or niching operator was needed to establish stable
and sustained sub-populations in various
regions of the Pareto-optimal front within
an evolving population. For two- or threeobjective optimization problems, once a
suitable niching is established, EA's other
operators are able to impose the needed
implicit parallelism to constitute an efficient search within each niche and also
among the niches.
In most problems, a low-dimensional
Pareto-optimal objective space comes
from a low-dimensional variable space
interactions, piece-wise or parametric,
despite having a large-dimensional variable space description of the original
problem [7]. However, when ManyObjective Problems (MaOP) involving
four or more objectives are to be solved,
variable interactions requiring to represent the Pareto-optimal solutions increase
and the simple implicit niching based
methods designed to solve two and
three-objective optimization problems
cease to find a diverse set of Pareto-optimal solutions on the entire front. A suitable normalization method for objectives
also becomes an important task [8]. The
need to preserve solutions from a widely
spread-out objective space for EA's
implicit parallelism mechanism to find
new and continuously evolving nondominated solutions gets less emphasized
by the so-called dominance-resistance
phenomenon. This fact has alluded EA
researchers for a long time and they took

a number of different paths to eventually
reliably solve MaOPs. A lot of efforts
have been made to modify the Pareto
dominance relationship, such as, with adominance [9], e-dominance [10], [11],
subspace dominance [12], fuzzy dominance [13], L-optimality [14], grid dominance [15], and preference order ranking
[16]. Another attempt to remedy the
dominance resistance phenomenon in
certain problems was to use a dimension
reduction technique [17]-[19], in which
redundant objectives were identified and
eliminated, as and when found, thereby
reducing the effective dimensionality of
the Pareto-optimal set. Other ways to
beat the high-dimensionality aspect were
to use a scalarized metric to provide
information about poorly progressed part
of the Pareto-optimal front and then
introduce an enhanced search. Indicatorbased EMO algorithms, such as hypervolume indicator [20]-[26], R2 indicator
[27], and I f + indicator [28] used a metric-based selection for solving MaOPs.
In 2007, Zhang and Li [29] came out
with a decomposition-based algorithm
(MOEA/D) that made the implicit niching method explicitly controlled, by
dividing the overall task of finding the
entire multi-dimensional Pareto-optimal
front into a number of loosely interacting
sub-tasks. MOEA/D relied more on
implicit parallelism to take place within
each sub-problem and leaving the automatic distribution of Pareto-optimal solutions to parallel but more independent
evolution of individual sub-problems.
The above discussion raises an
important distinction between the wellknown "implicit" parallelism associated
with a population-based EA and an
"explicit" control of it, introduced by
the algorithm developer. By designing
the operators, the developer controls the
resulting implicit parallelism induced in
the search process. For low-dimensional
objective space, no explicit control on
the extent of selection or recombination
operators was necessary. But for solving
large-dimensional objective space problems, an emphasis of any non-dominated
and less-crowded solution makes a too
generic search for an EMO algorithm to
converge to the entire Pareto-optimal

front effectively. An explicit control of
which population members should be
considered for each selection and recombination event is now necessary.
In this paper, we demonstrate the
need for such an explicit control in solving large-dimensional problems using
three main decomposition-based EMO
methods - MOEA/D, MOEA/D-M2M
and NSGA-III - for this purpose, but the
approach can be extended to other EA
and EMO methods as well. MOEA/
D-M2M [30], [31] is a new variant of
MOEA/D for population decomposition, and it decomposes an MOP/MaOP
into a set of multi- and many-objective
optimization subproblems. For ease of
distinction from MOEA/D, we call
MOEA/D-M2M as simply M2M in the
rest of this study. NSGA-III [32] is the
third generation non-dominated sorting
genetic algorithm, but it uses a decomposition-based niching method to achieve
the required convergence and diversity.
The remainder of this paper is organized as follows. Section II briefly introduces the basic working principles of
MOEA/D, M2M, and NSGA-III. Section III explains the concept of explicit
control on an algorithm's implicit parallelism and qualitatively discusses the
extent of explicit control introduced in
a number of existing EMO algorithms.
In Section IV, experiments are conducted with M2M algorithm and its variants.
The effect of explicit control of implicit
parallelism is demonstrated through simulation results on both unscaled and
scaled problems. Section V conducts
experimental studies and analyzes the
performance of two normalization procedures on MOEA/D and M2M variants. Further performance comparison
on a series of NSGA-III variants and
MOEA/D variants are conducted in
Section VI to investigate if the originally proposed versions could be improved
by using a better explicit control procedure. Interesting observations are made
in Section VII. Finally, Section VIII
concludes this paper.
II. Preliminaries

In this section, we br iefly introduce three main algorithms used in this

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