IEEE Computational Intelligence Magazine - November 2021 - 22

MFEARR [35], where the authors attempted to control the
computational resources by discarding the knowledge transfer
process according to the survival rate of cross-domain individuals.
MFEARR can perform well in both single-objective
and multi-objective multitasking problems, but its assumption
that is grounded on the original MFEA [13] is not applicable
to many state-of-the-art EMTO methods. Specifically, MFEARR
assumed when the parting way situation appears, during
which populations from varying domains lie in rather different
decision spaces, then information sharing among those
domains can hardly benefit the overall searching process.
Nevertheless, this assumption contradicts many emerging heterogeneous
domain adaptation algorithms [28, 36, 37] that
convert individuals explicitly from one domain to another.
MFEA/D-DRA [38] also resorted to allocating reasonably
distributed computational resources in EMTO-based MOO.
Instead of designing an allocation mechanism from scratch,
MFEA/D-DRA directly combined MOEA/D-DRA [39]
with the original MFEA [13], where the resource allocation
strategy simply resembled that of MOEA/D-DRA [39]. In
spite of the effectiveness, the resource allocation mechanism
in [38] highly relies on the MOEA/D framework, which is
inflexible since the allocation mechanism can only be applied
in MOEA/D based algorithms but cannot be utilized in
other MOO algorithms such as NSGA-II [6]. Different from
MFEA/D-DRA and MFEARR, Gong et al. [40] proposed a
flexible technique, called MTO-DRA, to capture the convergence
status of each optimization problem dynamically. In
MTO-DRA, the relative improvement of each optimization
task was recorded in a vector, and during each iteration, an
additional generation was assigned to a specific task based on
the softmax function of the vector. However, MTO-DRA has
several drawbacks. It cannot solve multitasking MOO problems
since the improvement vector is not applicable in the
context of MOO, it cannot adjust resource allocation intensity
and thus cannot flexibly cope with multitasking MOO
with various attributes, and its behavior has not been
explained theoretically or explicitly in the original paper [40].
Thus, according to the discussions above, it is of great necessity
to design a flexible resource allocation mechanism for multitasking
MOO problems.
To remedy the deficits aforementioned, in this article, we
propose a generalized resource allocation (GRA) framework
based on MTO-DRA [40], by extending MTO-DRA to
MOO problems and revealing its theoretical grounds explicitly.
It is notable that, although the proposed GRA can be viewed
as an extension of MTO-DRA, it can be applied to any multipopulation
based multitasking MOO algorithms. The contributions
of this paper are as follows:
❏ A normalized attainment function, as well as a multi-step
nonlinear regression, is designed for quantifying MOO convergence
performance, which extends MTO-DRA to
MOO problems.
❏ A generalized mathematical model is established for analyzing
the behavior and motivation of MTO-DRA, and by
22 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2021
this model, we propose a novel algorithmic procedure for
involving knowledge transfer information and adjusting
resource allocation intensity.
The remainder of this paper is organized as follows.
Section II analyzes MTO-DRA and Section III introduces the
proposed GRA framework. The effectiveness of the GRA
framework is justified by experimental analysis in Section IV,
and Section V concludes the paper.
II. Basics and Analysis on MTO-DRA
Since the generalized resource allocation algorithm proposed in
this paper can be considered as an extension of MTO-DRA
[40], the core component of MTO-DRA1 is introduced first in
Section II.A. Thereafter, the motivation and behavior of MTODRA
are analyzed from a theoretical perspective in Section
II.B, and hence its deficiencies and limitations can be pointed
out accordingly.
A. The Original MTO-DRA
In MTO-DRA, the multi-population framework is adopted
for EMTO problems, where different sub-populations are
intended to resolve different tasks. Under such a framework,
knowledge transfer among distinct tasks can be implemented
by the cross-population evolution process, where the i-th
population is incorporated with a random j-th population for
the subsequent evolution operations involving mutation and
crossover to optimize the i-th task. To distribute computational
resources towards these sub-populations, MTO-DRA
employs an index of improvement (IoI) vector for recording
the relative improvement of each sub-population within
given evolution iterations. To be specific, the relative improvement
value of sub-population i,
equation (3):
~ i can be formulated as
~ =
i
max fx tf xt T
fx tf xt T
|( ())( ())|
-{(
( )), (( -+n
ii
ii
where (( )),fx tTi
D
D
))}
(3)
D , and μ stand for the best fitness value for
problem i in generation round t, the evaluation interval to
measure the relative improvement, and a small floating number
to avoid division by zero, respectively. With these relative
improvement values, the IoI vector can be obtained by normalization
using softmax function that can be illustrated as
equation (4):
IoIi = /exp S ) ~j
j
exp S ) ~i
()
()
(4)
where IoIi, ~ i, and S represent the i-th item in the IoI vector,
the relative improvement value of the i-th task as calculated in
equation (3), and the coefficient of softmax function that is
1Although there is an adaptive control component for the random mating parameter
in MTO-DRA, this component is not considered in our paper because it has no
direct relationship with resource allocation.
IEEE Computational Intelligence Magazine - November 2021

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