IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 26, NO. 5, OCTOBER 2052
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Evolutionary Search With Multiview Prediction for
Dynamic Multiobjective Optimization
Zhang san , Li si
Abstract—Dynamic multiobjective optimization problem
(DMOP) denotes the multiobjective optimization problem
which varies over time. As changes in DMOP may exist some
patterns that are predictable, to solve DMOP, a number of
research efforts have been made to develop evolutionary search
with prediction approaches to estimate the changes of the
problem.
Index Terms—Dynamic multiobjective optimization, evolutionary search, prediction-based method.
I. Introduction
W
ULTIOBJECTIVE optimization problem (MOP)
considers the optimization of at least two conflicting objectives simultaneously [1][2].Unlike single objective
optimization, no single solution can satisfy a given MOP,
and the optima of a MOP are thus a set of solutions
in the decision space, namely, Pareto-optimal solutions
(POS)[3][4].
1) A multiview prediction method is proposed to predict
the changes of DMOPs from both the views of
decision and objective spaces, which could provide
more accurate guidance for evolutionary search in
solving DMOP.
2) An efficient kernelized autoencoding model is derived
to model the complex change patterns, which holds
a closed-form solution and will not bring much computational burden to the prediction.
3) The proposed method is designed to regenerate an
initial population for the evolutionary search when
dynamic occurs, which can be easily integrated into
existing MOEAs for solving DMOPs.
II. Preliminaries
In this section, we first give the brief introduction
of DMOPs. Subsequently, the brief review of existing
predictionbased evolutionary algorithms, and the motivation of our proposed evolutionary search with multiview
prediction for solving DMOP are presented.
A. Dynamic Multiobjective Optimization Problem
In this article, we consider the DMOP as a timevariant MOP. In particular, minimization problems are
investigated here. Formally, the mathematical definition
of a DMOP is formulated as
Manuscript received 17 March 2051; revised 20 July 2051 and 22
October 2051; accepted 4 December 2051. Date of publication 14
December 2051; date of current version 3 October 2052..
Algorithm 1 this is a algorithm
Require: n ≥ 0 ∨ x ̸= 0
Ensure: y = xn
y←1
if n < 0 then
X ← 1/x
N ← −n
else
X←x
N ←n
end if
while N ̸= 0 do
if N is even then
X ←X ×X
N ← N/2
else {N is odd}
y ←y×X
N ←N −1
end if
end while
TABLE I
OVERVIEW OF REPRESENTATIVE PREDICTION-BASED
DMOEAS.
algorithm
FPS
PPS
KF
main idea
using a forecasting techniques,
predicting the whole popluation
using a Kalman Filter-base model
remarks
1
2
3
B. Prediction-Based Evolutionary Algorithms for Dynamic Multiobjective Optimization
Generally, the workflow of existing prediction approaches can be summarized by Algorithm 1. In contrast
to the flow of static MOEAs, change detection and
population prediction are two additional components in
the prediction-based DMOEAs. In particular, the common
procedure of population prediction is as follows. First, a
learning model is constructed to learn the change patterns based on past search experience. Next, the learned
patterns are used to estimate subsequent change and to
predict the initial individuals after the dynamic change.
C. Motivation of Conducting Prediction From Multiple
Views
Real-world and benchmark DMOPs often possess dynamic changes with different patterns in POS and POF
IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 26, NO. 5, OCTOBER 2052
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TABLE II
OVERVIEW OF REPRESENTATIVE PREDICTION-BASED DMOEAS.
algorithm
MDP
PBDMO
SVR
main idea
a mutidirectional prediction strategy
a reaction mechanism that contains three different strategies
constructing a predictor with SVR
(i.e., dynamics in decision space and objective space). As
mentioned in Section II-A, both POS and POF change
over time in Type II DMOPs, and the change patterns
may be significantly different.
{
min
f1 (x) = g(x)(x1 + 0.1 sin(3πx1 ))α1
f2 (x) = g(x)(1 − x1 + 0.1 sin(3πx1 ))α2
(1)
x = t + cos t + 1
(2)
y = 2 sin t
(3)
III. PROPOSED METHOD
In this section, the details of the proposed evolutionary
search with multiview prediction for solving DMOPs
are presented. In particular, as illustrated in Fig. 2, by
employing a conventional MOEA as the basic optimization
solver, the search process starts as routine, following
the procedures as in the static environment. However,
an additional environmental-change detection method is
designed to trigger the population prediction, wherein a
prediction model is constructed to predict new individuals
for the subsequent search in the changed environments.
The search process will be iteratively performed until
predefined stop criteria are satisfied. To estimate dynamic
changes that occur in different spaces of the DMOP, as
shown in Fig. 3, we derive a kernelized autoencoding
model to conduct the population prediction from the
views of both decision and objective spaces. The input
and output of this model are the nondominated solutions
obtained along the search and the the newly predicted
population, respectively. In what follows, the dynamic
detection operator employed in the proposed method, the
details of the proposed kernelized autoencoding model for
multiview prediction, and the pseudocode of the proposed
multiview prediction for solving DMOP are presented.
A. Dynamic Detection
Existing dynamic detection methods can be categorized into two groups, i.e., detector-based detection and
behavior-based detection. In particular, the detector-based
methods reevaluate some specific solutions (detectors)
to detect changes in their function values or feasibility,
while the behavior-based approaches consider to assess
the behaviors of an algorithm for dynamic detection.
Following the robust detection performance achieved in
previous DMOEAs, to detect the dynamic changes, here
we randomly select 5% individuals as detectors in the
remarks
4
5
6
population. In particular, at the start of each generation,
the detectors will be reevaluated. If the current objective
values of these detectors vary from their stored ones in
the last generation, it is recognized that an environmental
change occurs.
B. Proposed Kernelized Autoencoding Model for Multiview Prediction
To learn diverse change patterns in DMOPs from
multiple views, in this article, we propose a kernelized
autoencoding model to track the moving of high-quality
solutions in both decision and objective spaces. In particular, in the context of evolutionary dynamic multiobjective
optimization, we denote the optimization problems of
a DMOP before and after the dynamic occurs as OP1
and OP2, respectively. Suppose two sets of solutions P
=p1,...,pN �Rd×N and Q =q1,...,qN �Rd×N, where N
denotes the number of solutions in each set and d is
the variable dimension are representative solutions of OP1
and OP2, respectively. By treating OP1 as the corrupted
version of OP2, the learning of change pattern from OP1 to
OP2 can then be modeled as the denoising autoencoding
process by configuring P and Q as the input and output of
the autoencoder, respectively, which is given by where M
� Rd×d is the learned mapping from OP1 to OP2, tr(A)
refers to the trace of the matrix A, and T is the transpose
operation of a matrix. denotes the mapped solutions of P
in an RKHS H that is transformed by a nonlinear mapping
function.
When a dynamic change of DMOP has been detected,
multiview prediction will be performed using the kernelized autoencoder derived above with the nondominated
solutions found along the evolutionary search process. In
particular, suppose the dynamic occurs at time instance
t+1, we propose to use the nondominated solutions obtained at time instance t and t−1, denoted by N DSt−1
and N DSt , respectively, to predict the new initial population for time instance t+1, from both decision and
objective spaces. First, half of the solutions in N DSt−1
S
and N DSt is randomly selected and stored in N DSt−1
S
and N DSt , respectively.
C. Pseudocode of the Proposed Multiview Prediction for
DMOP
Last but not the least, the combination of the solutions
predicted from the views of both decision and objective
spaces then composes the initial population IPt+1 as
change reaction in DMOEA, thereby to guide the search
toward the changed optima. It is straightforward to see
IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 26, NO. 5, OCTOBER 2052
that the proposed evolutionary search with multiview
prediction is optimizer independent, it thus can be easily
integrated into any of population-based static MOEAs for
tackling DMOPs.
D. Complexity Analysis
The computational cost of prediction component comes
from learning the mappings via kernelized autoencoding
model and solution prediction. The kernelized autoencoder
in Algorithm. 2 holds a closed-form solution, which is
obtained by solving the system of linear equations as
shown in (8), because the computational cost of solving
the system is dominated by the calculation of matrix
inverse, and the learning complexity of MS k and MF
k is O(d2 ∗ N) + O(m2 ∗ N) = O(d2 ∗ N), where N
denotes the number of nondominated solutions, d is the
variable dimension, and m is the number of objectives. In
solution prediction, the interior point algorithm calls for
the major time, which costs O(m3 ∗ N). Therefore, the
overall complexity of the proposed multiview prediction is
O((d2 + m3) ∗ N).
IV. EMPIRICAL STUDY
In this section, to evaluate the performance of the
proposed multiview prediction for solving DMOPs, comprehensive empirical studies on commonly used DMOP
benchmarks against four recently proposed DMOEAs are
presented.
A. Experimental Setup
1) Test Instances: The performance of the proposed
prediction approach is investigated on the recently
proposed DMOP benchmark for IEEE CEC2018
Competition on Dynamic Multiobjective Optimization. This test suite has 14 benchmark problems,
including nine two-objective and five three-objective
functions with diverse dynamic characteristics, such
as variable linkages, time-dependent POS/POF geometries, irregular POF shapes, disconnectivity, knee,
etc.
2) Performance Metrics: In the literature, there are
several metrics that have been proposed to evaluate
the performance of DMOEAs. In this article, we
employ two groups of commonly used metrics, i.e.,
the inverted generational distance (IGD) metric with
two dynamic variants, two dynamic variants of the
hypervolume (HV) metric, to objectively measure the
performan
3) Compared Algorithms: The proposed evolutionary
search with multiview prediction is compared against
four state-of-the-art prediction approaches and two
baselines with the separate components of our proposed method. In particular, the algorithms for comparison in the experiments are as follows.
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B. Results and Discussion
In this section, comparison results of the proposed
multiview prediction against three prediction-based approaches in terms of both optimization quality and efficiency are presented and discussed. Moreover, deeper
insights of the interaction between multiple views of
prediction are also investigated.
1) Comparison Against Existing Prediction-Based Approaches: The averaged MIGD and MHV values
with the standard deviations obtained by the compared algorithms based on MOEA/D optimizer under
five dynamic configurations (see Table II), over 20
independent runs on the IEEE CEC2018 DMOP
benchmarks, are presented in Tables III and IV.
The experimental results obtained by the algorithms
based on MOPSO optimizer are presented in Tables I
and II in the supplementary material. In these tables,
superior performances are highlighted in bold and
the Wilcoxon rank sum test at a 5% significance
level is performed on the results to indicate the
difference between the performances of baseline and
the proposed algorithm. “+,” “−,” and “�” denote
that the proposed MV-DMOEA is statistically significant better, worse, and similar to the compared
algorithm, respectively. In addition, we summarize
the results of the significance test with the obtained
MIGD and MHV values on 14 DMOP benchmarks
under five dynamic configurations (i.e., totally 70
DF instances) at the bottom row of each table.
w/t/l indicates that the proposed MV-DMOEA wins
on w instances, ties on t instances, and loses on l
instances, when compared to the algorithm with the
same optimizer, respectively. In the supplementary
material, we also calculate the effect size (scientific
significance) according to the Vargha–Delaney ˆ A12
statistic.
2) Discussion on Runtime of the Compared Algorithms:
It is worth noting that the prediction approaches
generally involve additional computational cost to
train the prediction model, such as the TCA-based
prediction model in and the SVR-based predictor.
Therefore, here we further compare and discuss the
runtime of all the compared algorithms under the
same implementation environment.
3) Comparisons of Prediction From Two Views: To
further investigate the deeper insights of the superior
performance obtained by the proposed multiview
prediction, we plot the initial solutions predicted from
the views of both decision and objective space in Fig.
6 and in the supplementary material. Here, we modify
the MV-MOEA/D to study prediction from each view
separately. Specifically, the kernelized autoencoder
model proposed in Section III-B is utilized to conduct
prediction from two different views. Fig. 6 depicts the
initial solutions predicted by the two independent
kernelized predictions on two two-objective and a
three-objective DMOPs at three time instances (i.e.,
IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 26, NO. 5, OCTOBER 2052
t = 15, 20, 25), respectively. As can be observed
in these figures, solutions obtained by the prediction
from the view of decision space (denoted by red circles
in the figures) are more diverse but relatively far
from the true POF, while the solutions predicted
from the view of objective space (denoted by black
circles in the figures) are lack of diversity but more
close to the true POF. The comparisons between
the prediction from the views of decision space and
objective space show that the prediction with learning
from multiple views could complement each other in
terms of both diversity and convergence. Therefore,
it is promising to combine the solutions estimated by
the two-view prediction, and to generate an initial
population as change reaction for accelerating the
evolutionary search toward the moving optima in
dynamic environments.
4) Sensitive Study of Parameter p in Multiview Prediction: In our proposed multiview prediction, the
size of the training set for each view is equally set,
whereas the proportion of either view may affect the
performance of the prediction approach. In order to
study how multiview prediction is sensitive to the
proportion, we set a hyperparameter p to denote
the proportion of predicted solutions from the view
of decision space in the whole initial population.
Specifically, we investigate five different p ranging
from 0.1 to 0.9, on four representative benchmarks
(i.e., DF1, DF9, DF10, and DF13, all of which possess
drastically different change patterns in decision and
objective space) with four different configurations (nt
= 1, 2.5, 5, 10,�t = 10) are chosen to test MVMOEA/D in multiview prediction. Fig. 7 shows the
averaged MIGD values obtained by MV-MOEA/D
on these test instances. The results indicate that
MV-MOEA/D performs relatively bad with too small
or too large value of p (i.e., p = 0.1, 0.9), while
the performance of the algorithm is obviously better
on these benchmarks when p is close to 0.5. From
these results, we can infer that both the prediction
in decision space and objective space is important
for predicting dynamic changes in solving DMOPs,
which again confirmed the motivation of the proposed
multiview prediction for solving DMOP.
V. REAL-WORLD CASE STUDY ON DYNAMIC
MULTIOBJECTIVE RECOMMENDER SYSTEM
Recommender systems are essential to cope with information overload in online services and have been widely
applied to various fields, such as e-commerce, movies,
streaming music, and personalized news reading. Typical
multiobjective recommendation aims to learn inherent
and long-term user preferences on different properties of
item,which are presumed to be static or slow varying
throughout the recommendation process. However, the
recommendation policy is generally modeled based on the
record of user’s historical behaviors, and her/his taste of
different items is actually changing over time. Thus, a
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fixed policy may lead to inappropriate recommendations
in the subsequent scenarios of the system. Bearing this
in mind, to demonstrate the effectiveness of the proposed
evolutionary search with multiview prediction in solving
real-world DMOP, in this section, we present a case study
with respect to the scenario of movie recommendation
and take into account user’s varying preferences on two
conflicting objectives of recommended movies.
VI. CONCLUSION
In this article, an evolutionary search with multiview
prediction has been proposed to solve DMOP. In particular, a kernelized autoencoding model has been derived to
perform the multiview prediction in an RKHS, which holds
a closedform solution and easy-to-solve. In the proposed
method, nondominated solutions found along the evolutionary search process have been used to build a prediction
model from the views of both decision and objective
spaces to generate an initial population, which serves as
the dynamic response that can be integrated into any
population-based multiobjective optimization algorithms
for solving DMOPs. To evaluate the performance of the
proposed multiview prediction, comprehensive empirical
studies have been conducted on the IEEE CEC2018
DMOP benchmarks under five different dynamic configurations. The results obtained by comparing against four
state-of-the-art prediction approaches and two baselines
with the separate components of multiview prediction
showed the efficacy of the proposed method for evolutionary dynamic multiobjective optimization. To further
demonstrate the application value of the proposed method,
a real-world DMOP of movie recommendation is formulated, and two traditional recommendation algorithms and
two DMOEAs are compared with the proposed multiview
prediction on the DMOP of movie recommendation. The
results of prediction accuracy verified that the multiview
prediction is able to solve real-world DMOPs.
The proposed multiview prediction has shown promising
performance on various benchmark problems and a practical problem; however, there are still several issues need to
be addressed to further improve the multiview prediction
for solving DMOP. First, the correlations between different views and how the dynamics in different spaces would
affect the changes of a DMOP are keen to the design
of multiview prediction for solving DMOP. Moreover,
the definition of knowledge that is used for prediction
when dynamic occurs may further affect the effectiveness
and efficiency of multiview prediction in DMOP. Finally,
constrained problems are commonly encountered in nature, and we did not make specific designs for constraints
except the boundary check in the prediction. Therefore,
for future work, we would like to consider the design of
the dynamic detection method to adaptively determine the
characteristics of dynamics, and suggest the appropriate
interaction strategy for the subsequent multiview prediction. In addition, we would like to further explore dynamic
multiobjective optimization with constraints considered in
the prediction.
IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 26, NO. 5, OCTOBER 2052
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Zhang san received the B.E. degree from the
Department of Computer Science, Anhui University, Hefei, China, in 2048, where he is
currently pursuing the Ph.D.degree.