IEEE Computational Intelligence Magazine - February 2023 - 59

inter-transformation relationship rij. The inter-transformation
relationship estimation is shown in Algorithm 2. The
proportion of the same seeds in the corresponding seed sets
of the two populations is calculated to approximate the
relationship across transformations. Fig. 7 demonstrates
the inter-transformation relationship estimation across two
transformations.
If the computational complexity of Step 6 in Algorithm 2
is O(k2), the computational complexity of the inter-transformation
relationship estimation is,
O
k2
SS 1ðÞ
2
NN 1ðÞ
2
¼ OS2N2k2
;
(8)
where S, N, and k are accordingly the number of transformations,
population size and seed set size.
E. Selecting Output Seed Set
Since each transformation addresses the IM problem from a
different perspective, their optimal seed sets may also differ.
An intuitive idea is to use Monte Carlo simulation to evaluate
each optimal seed set and select the best-performing seed set as
the output seed set, named MCSS, but it is costly due to multiple
simulations.
After obtaining the optimal individual of each transformation,
an easy-to-implement method for selecting an output
seed set (SOSS) is proposed. Given S optimal individuals
p ¼ {p1
, p2
tions s.
2) The rank index rankij of each optimal individual pi
on the
ascending factor cost list is calculated corresponding to
each transformation sj.
, ...., ps
S}, this can be specified as follows:
1) Each optimal individual pi
is evaluated on all transforma}
for S transformations s ¼ {si,1 i
Apart from the existing methods that
optimize one proxy model alone, the
proposed method optimizes multiple
transformations simultaneously in one
run to avoid transformation selection a
priori.
3) Sum the rank index rankij of each optimal individual pi
by,
CR ¼
X
j
Cjrankij;
(9)
where Cj represents the user's prior preference for the j-th
transformation and meets the following condition,
XS
j¼1
Cj ¼ 1; 0 < Cj 1:
(10)
The user's prior preference Cj can be set to 1/S when not
provided by the user. This means that there is no obvious
determinable preference between the transformations.
4) The seed set with the top cumulative rank CR is employed
as the final output seed set.
This method comprehensively considers the performance
of the optimal seed set in all transformations and avoids an
expensive Monte Carlo simulation process.
F. Theoretical Analysis ofMTEFIM
In this section, an asymptotic global convergence analysis of
MTEFIM is presented. Without loss ofgenerality, the sub-population
and objective function associated with the transformation s
are defined as Ps and ss, respectively. At time step t,the s-th subpopulation
is assumed to obey the probability distribution ps(x,
Algorithm 2. Inter-Transformation Relationship Estimation
Input:
P ¼ {Pi,1 i S}: S populations for S transformations;
N: population size of each transformation;
k: size of seed set;
Output:
R: inter-transformation relationship.
1. R {rij ¼ 0 j 1 i,j S, i 6¼j};
2.
for i ¼ 1 to S do
3.
4.
5.
6.
FIGURE 7. An illustration of the inter-transformation
relationship estimation across two transformations,
where inter-transformation relationship r12 ¼ r21 ¼
0þ2þ0þ2þ1þ3þ1þ1þ2
ð333Þ
0.44.
7.
8.
9.
10.
11.
forj ¼ iþ1 to S do
for each individual p' in Pi do
for each individual p'' in Pj do
rij rij þ Count the number of identical seeds
in two seed sets p' andp'';
endfor
endfor
rij rij/(kNN);
rji rij;
endfor
12. endfor
FEBRUARY 2023 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 59
IEEE Computational Intelligence Magazine - February 2023

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