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On the Combined Impact of Population Size and Sub-problem Selection in MOEA/D

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This paper intends to understand and to improve the working principle of decomposition-based multi-objective evolutionary algorithms. We review the design of the well-established Moea/d framework to support the smooth integration of different strategies for sub-problem selection, while emphasizing the role of the population size and of the number of offspring created at each generation. By conducting a comprehensive empirical analysis on a wide range of multi-and many-objective combinatorial NK landscapes, we provide new insights into the combined effect of those parameters on the anytime performance of the underlying search process. In particular, we show that even a simple random strategy selecting sub-problems at random outperforms existing sophisticated strategies. We also study the sensitivity of such strategies with respect to the ruggedness and the objective space dimension of the target problem.
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On the Combined Impact of Population
Size and Sub-problem Selection
in MOEA/D
Geoffrey Pruvost1(B
), Bilel Derbel1(B
), Arnaud Liefooghe1(B
),KeLi
2(B
),
and Qingfu Zhang3(B
)
1University of Lille, CRIStAL, Inria, Lille, France
{geoffrey.pruvost,bilel.derbel,arnaud.liefooghe}@univ-lille.fr
2University of Exeter, Exeter, UK
k.li@exeter.ac.uk
3City University Hong Kong, Kowloon Tong, Hong Kong
qingfu.zhang@cityu.edu.hk
Abstract. This paper intends to understand and to improve the work-
ing principle of decomposition-based multi-objective evolutionary algo-
rithms. We review the design of the well-established Moea/d framework
to support the smooth integration of different strategies for sub-problem
selection, while emphasizing the role of the population size and of the
number of offspring created at each generation. By conducting a compre-
hensive empirical analysis on a wide range of multi- and many-objective
combinatorial NK landscapes, we provide new insights into the combined
effect of those parameters on the anytime performance of the underlying
search process. In particular, we show that even a simple random strat-
egy selecting sub-problems at random outperforms existing sophisticated
strategies. We also study the sensitivity of such strategies with respect to
the ruggedness and the objective space dimension of the target problem.
1 Introduction
Context. Evolutionary multi-objective optimization (EMO) algorithms [7]have
been proved extremely effective in computing a high-quality approximation of
the Pareto set, i.e., the set of solutions providing the best trade-offs among the
objectives of a multi-objective combinatorial optimization problem (MCOP).
Since the working principle of an evolutionary algorithm (EA) is to evolve a
population of solutions, this population can be explicitly mapped with the tar-
get approximation set. The goal is then to improve the quality of the popula-
tion, and to guide its incumbent individuals to be as close and as diverse as
possible w.r.t. the (unknown) Pareto set. Existing EMO algorithms can be dis-
tinguished according to how the population is evolved. They are based on an
iterative process where at each iteration: (i) some individuals (parents) from
the population are selected, (ii) new individuals (offspring) are generated using
variation operators (e.g., mutation, crossover) applied to the selected parents,
c
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132 G. Pruvost et al.
and (iii) a replacement process updates the population with newly generated
individuals. Apart from the problem-dependent variation operators, the design
of selection and replacement is well-understood to be the main challenge for an
efficient and effective EMO algorithm, since these interdependent steps allow to
control both the convergence of the population and its diversity. In contrast with
dominance- (e.g., [7]) or indicator-based (e.g., [2]) approaches, aggregation-based
approaches [16] rely on the transformation of the objective values of a solution
into a scalar value, that can be used for selection and replacement. In this paper,
we are interested in studying the working principles of this class of algorithms,
while focusing on the so-called Moea/d (Multi-objective evolutionary algorithm
based on decomposition) [11,22], which can be considered as a state-of-the-art
framework.
Motivations. The Moea/d framework is based on the decomposition of the
original MCOP into a set of smaller sub-problems that are mapped to a popu-
lation of individuals. In its basic variant [22], Moea/d considers a set of single-
objective sub-problems defined using a scalarizing function transforming a multi-
dimensional objective vector into a scalar value w.r.t. one weight (or direction)
vector in the objective space. The population is then typically structured by
mapping one individual to one sub-problem targeting a different region of the
objective space. Individuals from the population are evolved following a coopera-
tive mechanism in order for each individual (i) to optimize its own sub-problem,
and also (ii) to help solving its neighboring sub-problems. The population hence
ends up having a good quality w.r.t. all sub-problems. Although being extremely
simple and flexible, the computational flow of Moea/d is constantly redesigned
to deal with different issues. Different Moea/d variants have been proposed so
far in the literature, e.g., to study the impact of elitist replacements [19], of
generational design [13], or of stable-matching based evolution [12], and other
mechanisms [1]. In this paper, we are interested in the interdependence between
the population size, which is implied by the number of sub-problems defined in
the initial decomposition, and the internal evolution mechanisms of Moea/d.
The population size has a deep impact on the dynamics and performance of
EAs. In Moea/d, the sub-problems target diversified and representative regions
of the Pareto front. They are usually defined to spread evenly in the objective
space. Depending on the shape of the (unknown) Pareto front, and on the num-
ber of objectives, one may need to define a different number of sub-problems.
Since, the population is structured following the so-defined sub-problems, it is
not clear how the robustness of the Moea/d selection and replacement strate-
gies can be impacted by a particular setting of the population size. Conversely,
it is not clear what population size shall be chosen, and how to design a selection
and replacement mechanism implying a high-quality approximation. Besides, the
proper setting of the population size (see e.g. [6,8,20]) in is EAs can depend on
the problem properties, for example in terms of solving difficulty. EMO algo-
rithms are no exceptions. In Moea/d, sub-problems may have different charac-
teristics, and the selection and replacement mechanisms can be guided by such
Population Size and Sub-problem Selection in MOEA/D 133
considerations. This is for example the case for a number of Moea/d variants
where it is argued that some sub-problems might be more difficult to solve than
others [18,23], and hence that the population shall be guided accordingly.
Methodology and Contribution. In this paper, we rely on the observation
that the guiding principle of Moea/d can be leveraged in order to support a
simple and high-level tunable design of the selection and replacement mecha-
nisms on one hand, while enabling a more fine-grained control over the choice
of the population size, and subsequently its impact on approximation quality on
the other hand. More specifically, our work can be summarized as follows:
We consider a revised design of Moea/d which explicitly dissociates between
three components: (i) the number of individuals selected at each generation,
(ii) the strategy adopted for selecting those individuals and (iii) the setting
of the population size. Although some sophisticated strategies to distribute
the computational effort of sub-problems exploration were integrated within
some Moea/d variants [10,18,23], to the best of our knowledge, the indi-
vidual impact of such components were loosely studied in the past.
Based on this fine-grained revised design, we conduct a comprehensive analy-
sis about the impact of those three components on the convergence profile of
Moea/d. Our analysis is conducted in an incremental manner, with the aim
of providing insights about the interdependence between those design com-
ponents. In particular, we show evidence that the number of sub-problems
selected at each generation plays an even more important role than the way
the sub-problems are selected. Sophisticated selection strategies from the lit-
erature are shown to be outperformed by simpler, well configured strategies.
– We consider a broad range of multi- and many-objective NK landscapes,
viewed as a standard and difficult family of MCOP benchmarks, which is
both scalable in the number of objectives and exposes a controllable difficulty
in terms of ruggedness. By a thorough benchmarking effort, we are then able
to better elicit the impact of the Moea/d population size, and the robustness
of selection strategies on the (anytime) approximation quality.
It is worth noticing that our work shall not be considered as yet another variant
in the Moea/d literature. In fact, our analysis precisely aims at enlightening the
main critical design parameters and components that can be hidden behind a suc-
cessful Moea/d setting. Our investigations are hence to be considered as a step
towards the establishment of a more advanced component-wise configuration
methodology allowing the setting up of future high-quality decomposition-based
EMO algorithms for both multi- and many-objective optimization.
Outline. In Sect. 2, we recall basic definitions and we detail the working princi-
ple of Moea/d. In Sect. 3, we describe our contribution in rethinking Moea/d
by explicitly dissociating between the population size and the number of selected
sub-problems, then allowing us to leverage existing algorithms as instances of
the revised framework. In Sect. 4, we present our experimental study and we
134 G. Pruvost et al.
state our main findings. In Sect. 5, we conclude the paper and discuss further
research.
2 Background
2.1 Multi-objective Combinatorial Optimization
Amulti-objective combinatorial optimization problem (MCOP) can be defined
by a set of Mobjective functions f=(f1,f
2,...,f
M), and a discrete set X
of feasible solutions in the decision space.LetZ=f(X)IR Mbe the set of
feasible outcome vectors in the objective space. To each solution xXis assigned
an objective vector zZ, on the basis of the vector function f:XZ.In
a maximization context, an objective vector zZis dominated by a vector
zZiff m∈{1,...,M},zmz
mand m∈{1,...,M}s.t. zm<z
m.A
solution xXis dominated by a solution xXiff f(x) is dominated by
f(x). A solution xXis Pareto optimal if there does not exist any other
solution xXsuch that xis dominated by x. The set of all Pareto optimal
solutions is the Pareto set. Its mapping in the objective space is the Pareto
front. The size of the Pareto set is typically exponential in the problem size. Our
goal is to identify a good Pareto set approximation, for which EMO algorithms
constitute a popular effective option [7]. As mentioned before, we are interested
in aggregation-based methods, and especially in the Moea/d framework which
is sketched below.
2.2 The Conventional MOEA/D Framework
Aggregation-based EMO algorithms seek good-performing solutions in multiple
regions of the Pareto front by decompo sing the original multi-objective problem
into a number of scalarized single-objective sub-problems [16]. In this paper, we
use the Chebyshev scalarizing function: g(x, ω) = maxi∈{1,...,M }ωi·
z
ifi(x)
,
where xX,ω=(ω1,...,ω
M) is a positive weight vector, and z=(z
1,...,z
M)
is a reference point such that z
i>f
i(x)xX,i∈{1,...,M}.
In Moea/d [22], sub-problems are optimized cooperatively by defining a
neighborhood relation between sub-problems. Given a set of μweight vec-
tors Wμ=(ω1,...,ω
μ), with ωj=(ωj
1,...,ω
j
M) for every j∈{1,...,μ},
defining μsub-problems, Moea/d maintains a population Pμ=(x1,...,x
μ)
where each individual xjcorresponds to one sub-problem. For each sub-problem
j∈{1,...,μ}, a set of neighbors Bjis defined by considering the Tclosest
weight vectors based on euclidean distance. All sub-problems are considered at
each generation. Given a sub-problem j, two sub-problems are selected at ran-
dom from Bj, and the two corresponding solutions are considered as parents.
An offspring xis created by means of variation (e.g., crossover, mutation). For
every k∈B
j,ifximproves k’s current solution xk, then xreplaces it, i.e.,
if g(x
k)<g(xj
k) then xk=x. The algorithm loops over sub-problems,
i.e., weight vectors, or equivalently over the individuals in the population, until
Population Size and Sub-problem Selection in MOEA/D 135
a stopping condition is satisfied. In the conventional Moea/d terminology, an
iteration refers to making selection, offspring generation, and replacement for one
sub-problem. By contrast, a generation consists in processing all sub-problems
once, i.e., after one generation μoffspring are generated. Notice that other issues
are also addressed, such as the update of the reference point zrequired by the
scalarizing function, and the option to incorporate an external archive for storing
all non-dominated points found so far during the search process.
From the previous description, it should be clear that, at each iteration,
Moea/d is applying an elitist (T+1)-EA w.r.t. the sub-population Biunderlying
the neighborhood of the current sub-problem. After one generation, one can
roughly view Moea/d as applying a (μ+μ)-EA w.r.t. the full population. A
noticeable difference is that the basic Moea/d is not a generational algorithm, in
the sense that it does not handle the population as a whole, but rather in a local
and greedy manner. This is actually a distinguishable feature of Moea/d, since
the population is structured by the initial sub-problems and evolved accordingly.
3 Revising and Leveraging the Design of MOEA/D
3.1 Positioning and Rationale
As in any EA, both the population size and the selection and replacement mech-
anisms of Moea/d play a crucially important role. Firstly, a number of weight,
i.e., a population size, that is too small may not only be insufficient to cover well
the whole Pareto front, but may also prevent the identification of high-quality
solutions for the defined sub-problems. This is because the generation of new off-
spring is guided by the so-implied (T+1)-EA for which the local sub-population
of size Tmight be too different and hence too restrictive for generating good
offspring. On the other hand, a too large population may result in a substantial
waste of resources, since too many sub-problems might map to the same solu-
tion. Secondly, a small population size can be sufficient to approach the Pareto
front in a reduced number of steps. However, a larger population is preferable
to better cover the Pareto front. As in single-objective optimization, a larger
population might also help escaping local optima [20]. As a result, it is not clear
what is a proper setting of the population size in Moea/d, since the previously
discussed issues seem contradictory.
Although one can find different studies dealing with the impact of the pop-
ulation size in EAs [5,6,8,20], this issue is explicitly studied only to a small
extent, especially for decomposition-based multi- and many-objective optimiza-
tion [9,15]. For instance, in [8], offline and online scheduling strategies for control-
ling the population size are coupled with SMS-EMOA [2], a well-known indicator-
based EMO algorithm, for bi-objective continuous benchmarks. Leveraging such
a study to combinatorial domains with more than two objectives, and within
the Moea/d framework, is however a difficult question. Tightly related to the
population size, other studies investigate the distribution of the computational
effort over the sub-problems [3,4,10,18,23]. The rationale is that the defined
sub-problems might have different degrees of difficulty and/or that the progress
136 G. Pruvost et al.
over some sub-problems might be more advanced than others in the course of
the search process. Hence, different adaptive mechanisms have been designed in
order to detect which sub-problems to consider, or equivalently which solutions
to select when generating a new offspring. A representative example of such
approaches is the so-called Moea/d–Dra (Moea/d with dynamical resource
allocation) [23], that can be considered as a state-of-the-art algorithm when deal-
ing with the proper distribution of the computational effort over sub-problems.
In Moea/d–Dra, a utility function is defined w.r.t. the current status of sub-
problems. A tournament selection is used to decide which sub-problems to select
when generating a new offspring. Despite a skillful design, such an approach
stays focused on the relative difficulty of solving sub-problems, while omitting
to analyze the impact of the number of selected sub-problems and its interaction
with both the population size and the characteristics of the underlying problem.
We propose to revise the Moea/d framework and to study in a more explicit
and systematic manner the combined effect of population size and sub-problem
selection in light of the properties of the MCOP at hand. As mentioned above,
this was investigated only to a small extent in the past although, as revealed
by our experimental findings, it is of critical importance to reach an optimal
performance when adopting the Moea/d framework.
3.2 The Proposed MOEA/D–(μ,λ,sps) Framework
In order to better study and analyze the combined effect of population size and
sub-problem selection, we propose to rely on a revised framework for Moea/d,
denoted Moea/d–(μ,λ,sps), as defined in the high-level template depicted in
Algorithm 1. This notation is inspired by the standard (μ+λ)-EA scheme, where
starting from a population of size μ,λnew individuals are generated and merged
to form a new population of size μafter replacement. In the Moea/d framework,
however, this has a specific meaning as detailed in the following.
The proposed Moea/d–(μ,λ,sps) algorithm follows the same steps as the
original Moea/d. However, it explicitly incorporates an additional component,
denoted sps, which stands for the sub-problem selection strategy. Initially, the
population is generated and mapped to the initial μweight vectors. An optional
external archive is also incorporated in the usual way with no effect on the search
process. The algorithm then proceeds in different generations (the outer while
loop). At each generation, λsub-problems, denoted Iλ, are selected using the
sps strategy. A broad range of deterministic and stochastic selection strategies
can be integrated. In particular, λcan be though as an intrinsic parameter of
the EMO algorithm itself, or implied by a specific sps strategy. The so-selected
sub-problems are processed in order to update the population (the inner for
loop). For the purpose of this paper, we adopt the same scheme than conven-
tional Moea/d: selected sub-problems are processed in an iterative manner,
although other generational EA schemes could be adopted. At each iteration,
that is for each selected sub-problem, denoted i, some parents are selected as
usual from the T-neighborhood Biof weight vector ωiw.r.t. Wμ. The setting of
the neighborhood Bican be exactly the same as in conventional Moea/d and
Population Size and Sub-problem Selection in MOEA/D 137
Algorithm 1. High level template of Moea/d–(μ,λ,sps)
Input: Wμ:= ω1,...,ω
μ: weights; g(·|ω): scalar function; T: neighb. size;
1EP ← : (optional) external archive ;
2Pμx1,...,x
μ: generate and evaluate initial population of size μ;
3zinitialize reference point from Pμ;
4while StoppingCriteria do
5Iλsps(Wμ,Pμ,history);
6for iIλdo
7Bithe T-neighborhood of sub-problem iusing Wμ;
8X←matingSelection(Bi);
9xvaria tion( X);
10 F(x)evaluate x;
11 EP ← update external archive using x;
12 zupdate reference point using F(x);
13 Pμreplacement(Pμ,x
,Bi|g);
14 history update search history;
its variants. However, at this step, it is important to emphasize that the consid-
ered neighborhood Biis w.r.t. the whole set of available weight vectors Wμ, that
is considering all the initially designed sub-problems, and not only the selected
ones. In particular, Bimay include some sub-problems that were not selected
by the sps strategy. This is motivated by the fact that parents that are likely to
produce a good offspring should be defined w.r.t. the population as a whole, and
not solely within the subset of active sub-problems at a given generation, which
might be restrictive. A new offspring xis then generated using standard vari-
ation operators (e.g., crossover, mutation). The reference point required by the
scalarizing function and the optional external archive are updated. Thereafter,
the offspring is considered for replacement as in the conventional Moea/d and
its variants. Here again, this is handled using the neighborhood Biof the current
sub-problem i, computed w.r.t. the whole population. It is worth noticing that
population update is made on the basis of the scalarizing function g, which is a
distinguishable feature of aggregation-based approaches.
At last, notice that we also use a history variable, referring to the evolution of
the search state, and hence serving as a memory where any relevant information
could be store for the future actions of the algorithm. In particular, we explicitly
integrate the history within the sps strategy, since this will allow us to leverage
some existing Moea/d variants, as further discussed below.
3.3 Discussion and Outlook
It shall be clear from the previous description that the Moea/d–(μ,λ,sps)
framework allows us to emphasize the interdependence between three main com-
ponents in a more fine-grained manner while following the same working principle
than the original Moea/d. Firstly, the number of weight vectors, or equivalently
138 G. Pruvost et al.
Table 1. Different instantiations of the Moea/d-(μ, λ, sps) framework.
Algorithm Pop. size # selected sub-prob. Selection strategy Ref.
Moea/d μ μ spsAll [22]
Moea/d–Dra μμ/5spsDra [23]
Moea/d–Rnd μ λ μspsRnd here
the population size, is now made more explicit. In fact, the set of weight vectors
now ‘simply’ plays the role of a global data structure to organize the individuals
from the population. This structure can be used at the selection and replacement
steps. In particular, one is not bound to iterate over all weight vectors, but might
instead select a subset of individuals following a particular strategy. Secondly, the
number of selected sub-problems λdetermines directly the number of offspring
to be generated at each generation. From an exploration/exploitation perspec-
tive, we believe this is of critical importance in general for (μ+λ)-EAs, and it
is now made more explicit within the Moea/d framework. Furthermore, the λ
offspring solutions are not simply generated from the individuals mapping to the
selected sub-problems. Instead, parent selection interacts directly with the whole
population, structured around the μweight vectors, since the local neighborhood
of each selected sub-problem may be used. Thirdly, the interaction between μ
and λis complemented more explicitly by the sub-problem selection strategy.
In conventional Moea/d for instance, the selection strategy turns out to be:
spsAll = ‘select all sub-problems’, with λ=μ. However, advanced Moea/d
variants can be captured as well. For instance, Moea/d–Dra [23], focusing
on the dynamic distribution of computations, can easily be instantiated as fol-
lows. For each sub-problem, we store and update the utility value as introduced
in [23] by using the history variable. Let us recall that in Moea/d–Dra,the
utility of a sub-problem is simply the amount of progress made by solution xi
for sub-problem ωiin terms of the scalarized fitness value g(·|ωi) over different
generations. In addition, Mboundary weight vectors (in the objective space) are
selected at each generation, and further (μ/5M) weight vectors are selected
by means of a tournament selection of size 10. Hence, the sub-problem selection
strategy turns out to be spsDra = ‘select the boundary vectors and sub-problems
using a tournament selection of size 10’, with λ=μ/5. Notice that this choice
is to recall the one-fifth success rule from (μ+λ) evolution strategies [14].
In the reminder, Moea/d–(μ,μ,spsAll) refers to the conventional Moea/d
as described in [22], and Moea/d–(μ,μ/5, spsDra) refers to Moea/d–Dra [23];
see Table 1. Other settings and parameters can be conveniently investigated as
well. Since we are interested in the combined effect of μ,λand sps, we also con-
sider a simple baseline sub-problem selection strategy, denoted spsRnd,whichis
to select a subset of sub-problems uniformly at random. Notice that our empir-
ical analysis shall shed more lights on the behavior and the accuracy of the
existing spsDra strategy.
Population Size and Sub-problem Selection in MOEA/D 139
4 Experimental Analysis
4.1 Experimental Setup
Multi-objective NK Landscapes. We consider multi-objective NK land-
scapes as a problem-independent model of multi-objective multi-modal combina-
torial optimization problems [17]. Solutions are binary strings of size Nand the
objective vector to be maximized is defined as f:{0,1}N→ [0,1]M. The param-
eter Kdefines the ruggedness of the problem, that is the number of (random)
variables that influence the contribution of a given variable to the objectives. By
increasing Kfrom0to(N1), problems can be gradually tuned from smooth
to rugged. We consider instances with the following settings: the problem size is
set to N= 100, the number of objectives to M∈{2,3,4,5}, and the ruggedness
to K∈{0,1,2,4}, that is, from linear to highly rugged landscapes. We generate
one instance at random for each combination.
Parameter Setting. For our analysis, we consider three competing algorithms
extracted from the Moea/d–(μ,λ, sps) framework as depicted in Table 1.For
the conventional Moea/d, only one parameter is kept free, that is the popu-
lation size μ.ForMoea/d–Dra, the sub-problem selection strategy is imple-
mented as described in the original paper [23]. We further consider to exper-
iment Moea/d–Dra with other λvalues. Recall that in the original vari-
ant, only μ/5 sub-problems are selected, while including systematically the
Mboundary weight vectors. For fairness, we follow the same principle when
implementing the Moea/d–Rnd strategy. Notice that the boundary weight
vectors were shown to impact the coordinates of the reference point zused
by the scalarizing function [18]. They are then important to consider at each
generation. To summarize, for both Moea/d–Dra and Moea/d–Rnd, two
parameters are kept free, namely the population size μand the number of
selected sub-problems λ. They are chosen to cover a broad range of values,
from very small to relatively very high, namely, μ∈{1,10,50,100,500}and
λ∈{1,2,5,10,25,50,100,150,200,300,400,450,500}such that λμ.
The other common parameters are set as follows. The initial weights are
generated using the methodology described in [21]. The neighborhood size is
set to 20% of the population size: T=0.2μ. Two parents are considered for
mating selection, i.e., the parent selection in the neighborhood of a current sub-
problem i. The first parent is the current solution xi, and the second one is
selected uniformly at random from Bi. Given that solutions are binary strings,
we use a two-point crossover operator and a bit-flip mutation operator where each
bit is flipped with a rate of 1/N .Moea/d–Dra involves additional parameters
which are set following the recommendations from [23].
Performance Evaluation. Given the large number of parameter values (more
than 2 000 different configurations), and in order to keep our experiments man-
ageable in a reasonable amount of time, every configuration is executed 10 inde-
pendent times, for a total of more than 20 000 runs. In order to appreciate the
140 G. Pruvost et al.
Fig. 1. Convergence profile of the conventional Moea/d w.r.t. population size (μ).
convergence profile and the anytime behavior of the competing algorithms, we
consider different stopping conditions of 100,101,...,107calls to the evalua-
tion function. Notice however that due to lack of space, we shall only report our
findings on the basis of a representative set of our experimental data.
For performance assessment, we use the hypervolume indicator (hv)[24]to
assess the quality of the obtained approximation sets, the reference point being
set to the origin. More particularly, we consider the hypervolume relative devi-
ation, computed as hvrd(A)=(hv(R)hv(A))/hv(R), where Ais the obtained
approximation set, and Ris the best Pareto front approximation, obtained by
aggregating the results over all executions and removing dominated points. As
such, a lower value is better. It is important to notice that we consider the exter-
nal archive, storing all non-dominated points found so far during the search pro-
cess, for performance assessment. This is particularly important when comparing
configurations using different population sizes.
4.2 Impact of the Population Size: spsAll with Varying μVal ues
We start our analysis by studying the impact of the population size for the con-
ventional Moea/d, that is Moea/d–(μ,μ,spsAll) following our terminology. In
Population Size and Sub-problem Selection in MOEA/D 141
Fig. 1, we show the convergence profile using different μvalues for the considered
instances. Recall that hypervolume is measured on the external archive.
For a fixed budget, a smaller population size allows the search process to focus
the computational effort on fewer sub-problems, hence approaching the Pareto
front more quickly. By contrast, using a larger population implies more diver-
sified solutions/sub-problems, and hence a better spreading along the Pareto
front. This is typically what we observe when a small and a large budget are
contrasted. In fact, a larger population size can be outperformed by a smaller one
for relatively small budgets, especially when the problem is quite smooth (K1)
and the number of objectives relatively high (M3). Notice also that it is not
straightforward to quantify what is meant by a ‘small’ population, depending on
the problem difficulty. For a linear bi-objective problem (M=2,K= 0), a par-
ticularly small population size of μ= 10 is sufficient to provide a relatively high
accuracy. However, for quadratic many-objective problems (M4, K= 1), a
small population size of μ=10(resp.μ= 50) is only effective up to a budget of
about 104(resp. 105) evaluations.
To summarize, it appears that the approximation quality depends both on
the problem characteristics and on the available budget. For small budgets, a
small population size is to be preferred. However, as the available budget grows,
and as the problem difficulty increases in terms of ruggedness and number of
objectives, a larger population performs better. These first observations suggest
that the anytime behavior of Moea/d can be improved by more advanced selec-
tion strategy, allowing to avoid wasting resources in processing a large number
of sub-problems at each iteration, as implied by the conventional spsAll strategy
which iterates over all sub-problems. This is further analyzed next.
4.3 Impact of the Sub-problem Selection Strategy
In order to fairly compare the different selection strategies, we analyze the impact
of λ, i.e., the number of selected sub-problems, independently for each strategy.
It is worth-noticing that both the value of λand the selection strategy impact
the probability of selecting a weigh vector. Our results are depicted in Fig. 2
for spsDra and spsRnd, for different budgets and on a representative subset of
instances. Other instances are not reported due to space restrictions. The main
observation is that the best setting for λdepends on the considered budget, on
the instance type, and on the sub-problem selection strategy itself.
Impact of λon spsRnd.For the random strategy spsRnd (Fig. 2, top), and for
smooth problems (K= 0), a small λvalue is found to perform better for a small
budget. As the available budget grows, the λvalue providing the best perfor-
mance starts to increase until it reaches the population size μ. In other words,
for small budgets one should select very few sub-problems at each generation,
whereas for large budgets selecting all sub-problems at each generation, as done
in the standard Moea/d, appears to be a more reasonable choice. However, this
tendency only holds for smooth many-objective problems. When the rugged-
ness increases, that is when the degree of non-linearity Kgrows, the effect of λ
142 G. Pruvost et al.
Fig. 2. Quality vs. number of selected sub-problems (λ) w.r.t. budget (μ= 500).
changes. For the highest value of K= 4, the smallest value of λ= 1 still appears
to be effective, independently of the available budget. However, the difference
with a large λvalue is seemingly less pronounced, especially for a relatively large
budget, and the effect of λseems to decay as the ruggedness increases. Notice
also that for the ‘easiest’ problem instance (with K=0andM= 2), it is only
for a small budget or for a high λvalue that we observe a loss in performance. We
attribute this to the fact that, when the problem is harder, search improvements
are scarce within all sub-problems, it thus makes no difference to select few or
many of them at each generation. By contrast, when the problem is easier, it is
enough to select fewer sub-problems, as a small number of improving offspring
solutions are likely sufficient to update the population.
Impact of λon spsDra.The impact of λappears to be different when analyzing
the spsDra strategy (Fig. 2, bottom). In fact, the effect of λseems relatively uni-
form, and its optimal setting less sensitive to the available budget and instance
type. More precisely, the smallest value of λ= 1 is always found to perform bet-
ter, while an increasing λvalue leads to a decrease in the overall approximation
quality. We attribute this to the adaptive nature of spsDra, for which the prob-
ability of selecting non-interesting sub-problems is smaller for lower λvalues.
Interestingly, in the original setting of Moea/d–Dra [23], from which spsDra
is extracted, the number of selected sub-problems is fixed to μ/5. Not only we
found that this setting can be sub-optimal, but it can actually be substantially
outperformed by a simple setting of λ=1.
spsAll vs. spsDra vs. spsRnd.Having gained insights about the effect of λfor the
different selection strategies, we can fairly analyze their relative performance by
using their respective optimal setting for λ. We actually show results with λ=1
for both spsDra and spsRnd. Although this setting was shown to be optimal for
spsDra, it only provides a reasonably good (but sub-optimal) performance in
the case of the simple random spsRnd strategy, for which other λvalues can
be even more efficient. Our results are shown in Fig.3for a subset of instances.
Population Size and Sub-problem Selection in MOEA/D 143
Fig. 3. Convergence profile of MOEA/D–(μ,λ,sps) w.r.t. sub-problem selection strat-
egy (μ= 500; λ= 500 for spsAll ,λ∈{1/5}for spsDra,andλ=1forspsRnd).
Table 2. Ranks and average hvrd value (between brackets, in percentage) obtained by
the different sps strategies after 104,105,106,107evaluations (a lower value is better).
Results for spsRnd and spsDra are for λ= 1. For each budget and instance, a rank of c
indicates that the corresponding strategy was found to be significantly outperformed
by cother strategies w.r.t. a Wilcoxon statistical test at a significance level of 0.05.
Ranks in bold correspond to approaches that are not significantly outperformed by any
other, and the underlined hvrd value corresponds to the best approach in average.
M K 104evaluations 105evaluations 106evaluations 107evaluations
spsAll spsDra spsRnd spsAll spsDra spsRnd spsAll spsDra spsRnd spsAll spsDra spsRnd
2 0 2(11.2) 1(10.5) 0(09.1) 2(09.4) 1(09.3) 0(09.0) 2(09.1) 0(09.0) 0(09.0) 0(09.0) 0(09.0) 2(09.0)
1 1(14.0) 1(13.2) 0(11.4) 1(10.9) 1(10.5) 0(09.5) 2(09.8) 1(09.6) 0(09.2) 2(09.5) 1(09.4) 0(09.2)
2 1(17.2) 0(15.6) 0(15.4) 1(13.0) 0(12.5) 0(11.7) 0(11.3) 0(10.9) 0(10.6) 0(10.2) 0(10.1) 0(09.8)
4 2(22.1) 0(19.5) 0(18.9) 1(17.5) 0(14.9) 0(16.0) 2(14.6) 0(13.3) 0(13.0) 0(13.1) 0(12.0) 0(12.2)
3 0 1(15.1) 1(15.0) 0(10.2) 1(11.0) 1(10.9) 0(09.2) 0(09.1) 0(09.1) 0(09.1) 0(09.0) 0(09.0) 2(09.1)
1 1(18.4) 1(18.4) 0(14.2) 1(13.3) 1(13.0) 0(10.5) 1(10.8) 1(10.5) 0(09.4) 1(09.5) 1(09.5) 0(09.0)
2 1(24.4) 1(23.4) 0(20.6) 1(16.3) 0(16.1) 0(14.7) 2(12.8) 1(12.3) 0(11.1) 1(11.7) 1(11.1) 0(09.4)
4 1(31.0) 0(29.9) 0(28.0) 0(22.7) 0(20.5) 0(21.4) 0(16.7) 0(15.4) 0(15.6) 0(13.6) 0(13.0) 0(12.2)
4 0 1(20.5) 1(20.5) 0(13.2) 1(12.9) 1(12.8) 0(09.9) 0(09.4) 0(09.4) 0(09.3) 0(09.0) 0(09.0) 2(09.2)
1 1(25.0) 1(24.9) 0(18.5) 1(15.2) 1(15.4) 0(11.8) 1(09.9) 1(09.8) 0(08.8) 1(08.4) 1(08.5) 0(07.9)
2 1(29.8) 1(30.6) 0(24.7) 1(19.5) 1(18.7) 0(16.0) 1(12.9) 1(12.3) 0(10.1) 1(09.8) 1(09.9) 0(07.9)
4 1(38.0) 1(37.1) 0(32.5) 1(26.4) 1(25.4) 0(22.5) 1(16.8) 1(16.6) 0(15.1) 0(11.6) 0(11.3) 0(10.5)
5 0 2(26.3) 1(25.2) 0(15.4) 1(14.1) 1(14.7) 0(10.2) 0(08.0) 1(08.3) 2(08.5) 0(07.4) 0(07.5) 2(08.0)
1 1(29.9) 1(29.9) 0(21.5) 1(16.5) 1(17.1) 0(13.0) 1(08.3) 1(08.5) 0(07.7) 0(05.9) 0(06.1) 1(06.1)
2 1(35.2) 1(34.1) 0(28.1) 1(21.4) 0(20.0) 0(17.6) 0(10.9) 0(10.5) 0(09.7) 0(06.8) 0(06.2) 0(05.3)
4 1(41.6) 1(40.7) 0(35.5) 1(26.5) 1(26.9) 0(24.1) 0(14.7) 0(15.1) 0(14.3) 0(06.0) 0(07.4) 0(07.4)
The spsAll strategy, corresponding to the conventional Moea/d [22], and spsDra
with λ=μ/5, corresponding to Moea/d–Dra [23], are also included. We can
see that the simple random selection strategy spsRnd has a substantially better
anytime behavior. In other words, selecting a single sub-problem at random is
likely to enable identifying a high-quality approximation set more quickly, for a
wide range of budgets, and independently of the instance type.
Pushing our analysis further, the only situation where a simple random strat-
egy is outperformed by the conventional Moea/d or by a Moea/d–Dra setting
144 G. Pruvost et al.
Fig. 4. Convergence profile of MOEA/D–(μ,1,spsRnd ) w.r.t. population size (μ).
using an optimal λvalue is essentially for the very highest budget (107evalu-
ations) and when the problem is particularly smooth (K= 0). This can be
more clearly observed in Table 2, where the relative approximation quality of
the different strategies are statistically compared for different budgets. Remem-
ber however that these results are for λ= 1, which is shown to be an optimal
setting for spsDra, but not necessarily for spsRnd where higher λvalues perform
better.
4.4 Robustness of MOEA/D–(μ,λ,spsRnd) w.r.t. μand λ
In the previous section, the population size was fixed to the highest value of
μ= 500. However, we have shown in Sect.4.2 that the anytime behavior of the
conventional Moea/d can be relatively sensitive to the setting of μ, in particular
for some instance types. Hence, we complement our analysis by studying the sen-
sitivity of the spsRnd strategy, which was found to have the best anytime behavior
overall, w.r.t the population size μ. Results for spsRnd with λ= 1 are reported
in Fig. 4. In contrast with the spsAll strategy from the conventional Moea/d
reported in Fig. 1, we can clearly see that the anytime behavior underlying spsRnd
is much more stable. In fact, the hypervolume increases with μ, independently
of the considered budget and instance type. Notice also that when using small
μvalues, convergence occurs much faster for smooth problems (K= 0) com-
pared against rugged ones (K= 4). This means that a larger population size μ,
combined with a small value of λ, shall be preferred.
From a more general perspective, this observation is quite insightful since
it indicates that, by increasing the number of weight vectors, one can allow
for a high-level structure of the population, being eventually very large. Notice
also that such a data structure can be maintained very efficiently in terms of
CPU time complexity, given the scalar nature of Moea/d. This is to contrast
with, e.g., dominance-based EMO algorithms, where maintaining a large popu-
lation may be computationally intensive, particularly for many-objective prob-
lems. Having such an efficient structure, the issue turns out to select some sub-
problems from which the (large) population is updated. A random strategy for
sub-problem selection is found to work arguably well. However, in order to reach
Population Size and Sub-problem Selection in MOEA/D 145
an optimal performance, setting up the number of sub-problems λmight require
further configuration issues. Overall, our analysis, reveals that a small λvalue,
typically ranging from 1 to 10, is recommended for relatively rugged problems,
whereas a large value of λshould be preferred for smoother problems.
5 Conclusions and Perspectives
In this paper, we reviewed the design principles of the Moea/d framework by
providing a high-level, but more precise, reformulation taking inspiration from
the (μ+λ) scheme from evolutionary computation. We analyzed the role of three
design components: the population size (μ), the number of sub-problems selected
(and then the number of offspring generated) at each generation (λ), and the
strategy used for sub-problems selection (sps). Besides systematically informing
about the combined effect of these components on the performance profile of
the search process as a function of problem difficulty in terms of ruggedness and
objective space dimension, our analysis opens new challenging questions on the
design and practice of decomposition-based EMO algorithms.
Although we are now able to derive a parameter setting recommendation
according to the general properties of the problem at hand, such properties
might not always be known beforehand by the practitioner, and other properties
might be considered as well. For instance, one obvious perspective would be to
extend our analysis to the continuous domain. More importantly, an interest-
ing research line would be to infer the induced landscape properties in order to
learn the ‘best’ parameter setting, either off-line or on-line; i.e. before or during
the search process. This would not only avoid the need of additional algorithm
configuration (tuning) efforts, but it could also lead to an even better anytime
behavior. One might for instance consider an adaptive setting where the values
of μ,λ,andsps are adjusted according to the search behavior observed over
different generations. Similarly, we believe that considering a problem where the
objectives expose some degree of heterogeneity, e.g., in terms of solving difficulty,
is worth investigating. In such a scenario, the design of an accurate sps strategy
is certainly a key issue. More generally, we advocate for a more systematic anal-
ysis of such considerations for improving our fundamental understanding of the
design issues behind Moea/d and EMO algorithms in general, of the key differ-
ences between EMO algorithm classes, and of their success in solving challenging
multi- and many-objective optimization problems.
Acknowledgments. This work was supported by the French national research agency
(ANR-16-CE23-0013-01) and the Research Grants Council of Hong Kong (RGC Project
No. A-CityU101/16).
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