IEEE Computational Intelligence Magazine - November 2021 - 26

The proposed GRA incorporates the knowledge
transfer information during the evolution process
and the resource allocation process.
parameter matrix rmp is maintained for capturing the knowledge
transfer information, where rmp ,ii
ty of knowledge transfer for task i, and rmp ,ij
indicates the probabiliindicates
the
chance that the j-th task can be involved in the evolution process
of the i-th task. Herein, the rmp matrix is dynamically
updated in response to the evolution status, as illustrated in
line 8 to line 12 of Algorithm 3. In detail, if the selected task i
can be improved by the task j during cross-population evolution,
then rmp ,ij
is enlarged by a knowledge transfer rate ktr, or
otherwise rmp ,ij is compressed by ktr. Consequently, this
dynamically updated rmp matrix can serve as the knowledge
that illustrates the relationship among the optimized tasks,
wherein larger rmp ,ij
indicates higher importance of the task j
to the task i, and vice versa.
Given the knowledge transfer information stored in the rmp
matrix, it can be employed as the prior information during
each resource allocation phase. Recall that MTO-DRA aims to
tackle a special case of optimization problem indicated in
equation (10), differently, GRA is able to handle this generalized
optimization problem in a more natural approach. Specifically,
to instantiate the generalized implicit objective function
indicated in equation (10), we substitute the importance factor
a k
by equation (11):
a = 1 / rmpk i
Z 1
K
k
i =
ik st
, ,
..
!
(11)
where Z is a normalization factor. Herein, ka
can indicate the importance of optimization
task k towards other tasks, and thus can serve as
the prior information for quantifying different
levels of importance for all the tasks. To apply
this prior information, instead of employing IoI
vector to allocate resources directly as in Algorithm 1, vector a
should also be involved, which will be discussed in Algorithm 5
of Section III.D at length.
D. MOO Extension in GRA
In MTO-DRA, the IoI vector can only be applied in singleobjective
optimization. Conversely, in GRA, we extend the IoI
vector for MOO problems by introducing a normalized attainment
function, and employ MOO-aware IoI vector to resolve
the generalized implicit objective function (10), where the
transfer information mentioned in Section III.C is involved.
1) Attainment Function
As discussed in the proposed Lemma, IoI vector is obtained by
softmax normalization of relative improvement value,
~ , which
measures the changing tendency of the fitness function. Fitness
function measurement can be directly attained in single-objective
optimization, but it is not trivial for MOO problems. Among
the metrics for quantifying the MOO performance, inverted
generational distance (IGD) and hypervolume (HV) are the most
prevalent ones [41]. However, IGD requires a preset reference set,
and HV calls for a preset reference point, which is not suitable
for online MOO performance measurement due to the handcrafted
settings. Therefore, the attainment function [42] is adopted
for evaluating MOO solution quality online. Given a set S,
attainment function is defined as equation (12):
Attain () = '
S v
f1
1
S1
S2
S3
where xi
1
0 otherwise
,
,
if # S
v
(12)
where v # S means that solution v dominates at least one of
the elements in set S. To evaluate the convergence performance
of the MOO solution set S, attainment function is employed for
spanning all the feasible solution, as illustrated in equation (13):
IS
() = #
and xi
(, ,...,)
(, ,...,)
xx x
xx x
12
12
n
indicate the lower bound and the upper
bound of the i-th dimension of the objective space, respectively.
To illustrate, a two-dimensional example is given in Fig.2,
wherein the range of solution space is [, ]01 2
and each point in
the grey area can dominate at least one solution in the solution
set { ,,
f2
1
FIGURE 2 Illustration of attainment function for a given set { ,,}
and .f2
SS S123
in the objective space. The objective space includes the objective
functions f1
one element in the set.
The points in the grey area can dominate at least
SS S12 3 }. Notably, this attainment function indicator
only measures the convergence performance of the MOO
solution set, but cannot consider diversity information. This is
because, the major concern is to allocate resources so that all
the tasks can converge in a limited computational budget,
wherein diversity is not taken into account. Although sharing
similar characteristics with the attainment function, HV calls
for hand-crafted settings such as reference points. Moreover, in
26 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2021
n
Attain ()d
S vv
(13)
IEEE Computational Intelligence Magazine - November 2021

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