IEEE Computational Intelligence Magazine - November 2021 - 24

Remark 1. In the analysis of Lemma, we consider the
weight factor as a small discrete value, D T. Notice that, in
K
min
Dt
MTO-DRA, i~ would undergo softmax function for normalization
as illustrated in equation (4), in which the softmax function
coefficient S can be accounted as a part of the weight factor since
it is the same for all the tasks. Hence, the overall weight factor
is STD.
Theorem.
Under the assumption that the current best
function is a convex function, the implicit objective function that
the resource allocation component of MTO-DRA aims to tackle
is as follows:
K
min gt t
tG
Dt
..
st DD
k=1
/
/
k
k ! 012 , ,...,
DD ],tG kK
[,
gtk
=
=
(9)
where () is current best fitness for optimization task k given t generation
rounds and tk is the generation rounds that have already been
exhausted in task k.
The readers can refer to Section S.II in the supplementary
material for the proof of this theorem. To dive deeper into this
theorem, we point out that MTO-DRA basically ignores the
prior information about the knowledge transfer process in
EMTO. To be specific, in a generalized perspective, the original
problem in equation (9) can be treated as a special case of optimization
problem stated as follows:
Algorithm 2 Algorithm Framework of GRA.
Data: Task size K, Population size N, Resource allocation probability
RAP, Index of improvement vector IoI, Time interval GD
Result: Best solution for each optimization task
1 /*Initialization*/
2 foreach task i do
3
4
5
IoI 1i
Initialize N/K individuals in i-th sub-population
Evaluate the individuals for task i
!
6 end
7 while termination condition is not met do
8
9
/*Generalized resource allocation*/
for i 1! to KG) D do
10
11
12
13
14
15
16
17
18
19
20
21
22
23 end
j 0!
if rand (, ) RAP then
/*Allocate resources*/
01 1
j ! Roulette-Wheel Selection on IoI vector
else
/*Normal evaluation*/
ji %K!
end
/*Refer to Algorithm 3*/
Evolve the j-th sub-population for task j
end
/*Refer to Algorithm 5*/
Re-evaluate improvement vector IoI
kk +D
k=1
K
()k
..
st DDk =
k=1
DD ],
k ! 012
[,
tG kK
101
/
/
/
K
k=1
aa ! [, ]
= , ,...,
k = ,
k
(10)
where ka serves as the importance indicator for each optimization
task k, which can be treated as prior information or problemdependent
knowledge. During the concurrent optimization process
in EMTO, there can be specific tasks that progress slowly but can
benefit the searching process of a vast number of other tasks, and
thus these tasks should be imposed on higher priorities. However,
MTO-DRA abandons such prior information, and treats all the
tasks equally in the implicit objective function (9), which remains
to be improved by incorporating knowledge transfer information.
III. The Proposed Generalized Resource
Allocation Algorithm
This section illustrates the proposed GRA algorithm at length.
Recall that this GRA algorithm can be regarded as an extension
of MTO-DRA [40], aiming at addressing the weakness
pointed out in Section II.B, Section III.A lists the major deficiencies
of MTO-DRA and the general framework of the
proposed GRA. More specifically, Section III.B describes the
controllable resource allocation intensity component and Section
III.C describes the incorporation of knowledge transfer
information. Section III.D introduces how to extend our
GRA framework into MOO settings in detail, followed by the
implementation details covered in Section III.E.
A. General Algorithm Framework of GRA
Based on the discussions in Section II.B, it can be seen that the
original resource allocation strategy in MTO-DRA has the
following deficiencies:
❏ It cannot flexibly adjust the resource allocation intensity
according to equation (8).
❏ It takes each optimization task equally regardless of prior
information, according to its implicit objective function in
equation (9).
❏ It cannot resolve MOO problems, since the IoI vector is not
able to naturally indicate the improvement of multiple
objective functions by a single improvement indicator i~ as
in equation (3).
To redeem the defects stated above, a generalized resource
allocation algorithm is put forward as an extension grounded on
MTO-DRA, as illustrated in Algorithm 2 and Fig.1. Similar
with MTO-DRA, the general idea of GRA is also allocating
more computational resources to the tasks having larger performance
improvements, as depicted in Fig.1. In Fig.1, the EMTO
framework mainly undergoes two steps: one is the population
evolution process, and the other is the resource allocation process.
In the population evolution process, an external performance
indicator that is introduced in subsection D can measure
24 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2021
a )gt t ()kk kk
+D
k=1
K
tG
IEEE Computational Intelligence Magazine - November 2021

Table of Contents for the Digital Edition of IEEE Computational Intelligence Magazine - November 2021
