IEEE Computational Intelligence Magazine - November 2021 - 41

C. Evolutionary Multi-Tasking Optimization
EMTO solves multiple optimization tasks
simultaneously in the EA framework, with the
aim of improving the solving of every single
task via inter-task knowledge transfer. Given
K optimization tasks {( ), ,( )}FF
with ():,F xR where kK1 f=
and X 3k R ,nk
11
kk
X " mk
xxKK
,,
f
the goal of EMTO is to identify
{, ,}
1
K
where xRk
) !3kX
xx argmin{( ), ,( )},KKFF
(, ,)1 f
gf=
))
11
xxK
nk
is the optimal solution of ()F xkk
xx
.
MFEA [11] is one of the most widely used EMTO algorithms.
It is extended from traditional EAs by introducing a set
of newly defined individual properties to assign rank-based
individual fitness together with four key components, i.e., unified
representation, assortative mating, vertical cultural transmission,
and selective evaluation, in order to enable knowledge
transfer between tasks. In particular, the following new individual
properties were introduced in MFEA:
Factorial rank: The factorial rank rik
on task k is the rank of xi
of an individual xi
in the population sorted in terms of
the k-th task.
Skill factor: The skill factor ix of an individual xi
task on which the factorial rank of xi
x = argmin {}.r
iik
1 kK
##
=
is the factorial rank of xi
is the
is the smallest, i.e.,
Scalar fitness: the scalar fitness ih of an individual xi
denoted as hii1 /r ix where ri ix
(3)
is
on its
skill factor task.
The unified representation is proposed to overcome the issue
of " the curse of dimensionality " and facilitate cross-task knowledge
transfer. All individuals are encoded into a unified search
space Ω by integrating
XX The dimensionality n of Ω is
1 ,...,.K
defined as the maximum dimensionality of the elemental subspaces,
i.e.,
= max {,...,
1
nn nK}. The first nk dimensions of an
individual in the unified space are then used to decode the corresponding
representation in the k-th task. In assortative mating,
individuals from the same task are assigned with a high probability
to mate for generating offspring, whereas individuals from
different tasks mate at a lower probability. In vertical cultural
transmission, an offspring individual randomly inherits the skill
factor from one of its parents, leading to individual exchange
across the tasks. Finally, selective evaluation is proposed to
reduce resource consumption, that is, each offspring evaluates
only on one task associated with its skill factor.
Numerous attempts have been made to improve the performance
of MFEA. For example, new types of knowledge,
such as search direction [19]-[21], best solution [33], and
population distribution [29], are abstracted and transferred
among the tasks, which largely enhances the positive intertask
knowledge transfer. Improvement can also be achieved
via adaptive mechanisms of " explicit transfer, " i.e., learning
the relationship among component
tasks and transferring
EMTO solves multiple optimization tasks
simultaneously in the EA framework, with the
aim of improving the solving of every single task
via inter-task knowledge transfer.
such information among tasks, e.g., linear transformation
[22]-[24] and nonlinear transformation [34]. These two
explicit transfers manage to improve population diversity. The
frequency of knowledge transfer is a key factor in the performance
of MFEAs. Wen and Ting [35] adjusted the frequency
of knowledge transfer according to its probability of producing
better solutions. Zheng et al. [36] introduced an ability
vector to capture the inter-task correlations and change the
intensity of knowledge transfer. Bali et al. [28] further proposed
MFEA-II by updating the frequency of knowledge
transfer based on online learning and inter-task similarities.
For expensive optimization, a target task can be better optimized
together with auxiliary tasks constructed by surrogate
models, such as polynomial regression [37], radial basis function
network [38], and Gaussian process [37]. The surrogate
models are built based on the historical search information of
the optimization task and can reduce the number of fitness
evaluations. MFEA has been extended to handle multiobjective
MTO [13] and many-task optimization [39], [40] by
improving the scalability. Different metaheuristics such as OBL
[41], differential evolution (DE) [42], and particle swarm optimization
(PSO) [43] were also integrated into MFEA to create
different variants.
III. Proposed Algorithm
In this section, we propose a new MFEA, namely MFEAGSMT,
by integrating an inter-task gene-similarity-based
knowledge transfer strategy and an adaptive mirror transformation
strategy. The framework of the proposed MFEAGSMT
is outlined in Algorithm 1. In the initialization (lines
1-3), a population P of N solutions is randomly generated,
evaluated, and assigned with their corresponding skill factors.
In the following evolution procedure (lines 5-32), the two
new strategies are used to improve the effectiveness of
knowledge transfer and search efficiency. Particularly, in
inter-task knowledge transfer, Gaussian distribution is introduced
to feature the probability density of each gene and
KLD [30] is employed to measure the distribution similarity
of the inter-task gene pairs (line 6). A selective crossover
based on the inter-task gene similarity is then used to reproduce
offspring solutions (lines 8-11). Given two parent solutions,
if they have the same skill factor, the selective crossover
is carried out to generate two offspring solutions; otherwise,
the crossover is performed under a predefined random mating
probability (rmp). The proposed inter-task knowledge
transfer is based on online gene similarity, instead of individual
similarity, to facilitate positive knowledge transfer. After
selective crossover, the adaptive mirror transformation based
NOVEMBER 2021 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 41
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