IEEE Computational Intelligence Magazine - February 2023 - 56

FIGURE 2 The core idea of MTFO, where multiple alternate transformations of a target task are optimized by multi-task EA simultaneously.
where D is the search space for the target optimization task f
and m is the dimension ofx. The purpose ofMTFO is to find
the best solutions to an optimization task f through effective
knowledge transfer across alternate formulations. This solution
can be selected from the following set,
X ¼ x x ¼ arg minfi xðÞ
; i ¼ 1; 2; :::; S :
(5)
In general, the MTFO problem is regarded as an MTO
problem. Evolutionary multi-task optimization (EMTO) aims
to solve multiple tasks simultaneously by utilizing some paradigms
motivated by EC [49], [50]. Gupta et al. [33] designed the
multifactorial EA (MFEA) to optimize many tasks at the same
time, inspired by cultural exchanges between organisms. MFEA
encodes individuals into a unified search space to provide a basic
guarantee for knowledge transfer and sets a skill factor for each
individual to indicate its associated task. By assortative mating
and selective imitation, knowledge is implicitly transferred
across tasks with a certain probability rmp to improve the convergence
performance. A theoretical analysis ofMFEA demonstrates
that knowledge transfer across tasks is controlled by rmp.
Bail et al. [36] further proposed MFEAII, based on online
parameter estimation, to address the prevalence of negative
transfers. An explicit EMTO algorithm, called EMEA in this
article, was proposed by Feng et al. [34] to utilize multiple evolutionary
solvers with different biases. In EMEA, each task is
assigned an evolutionary solver, and the top individuals on each
task are explicitly transferred to other tasks based on an autoencoding
that learns optimal linear mapping across tasks. Liaw
et al. [35] proposed symbiosis in biocoenosis (SB) optimization
to deal with MTO, which leverages the concept ofSB to transfer
knowledge across tasks. To the best of our knowledge,
although a series of excellent EMTO methods have been presented,
none is specifically designed to address the IM problem.
Fig. 2 illustrates the core idea of the MTFO, where multiple
alternate transformations ofa target task are optimized by a multitask
EA at thesametime. Theuniqueadvantagesand similarities
of the alternative formulas are exploited through the knowledge
transfer that occurs in the search process. The basic idea ofMTFO
56 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | FEBRUARY 2023
is employed to solve complex optimization problems in existing
works [38], [40], [41]. Ding et al. [40] proposed an effective
knowledge transfer across low-fidelity and high-fidelity transformations
for a single expensive problem. To solve large-scale optimization
problems, existing works use dimensionality reduction
Algorithm 1. Framework of MTEFIM
Input:
s ¼ {si,1 i S}: S proxy models (transformations) for
IM under an influence spread model;
N: population size of each transformation;
k: size of seed set;
MFE: maximum number of function evaluations;
Output:
A: output solution for IM.
1. P ¼ {Pi,1 i S} Initialize S populations;
2. fP ¼ {fPi,1 i S} Evaluate S populations on their
transformations;
3. while (stopping criterion is not met) do
4. R Learn the inter-transformation relationship according
to P (see Algorithm 2);
5.
for i ¼ 1 to S do
6. Oi Generate offspring based on designed genetic
operator;
7. endfor;
8.
9.
10.
11.
12.
13.
14.
15.
16.
for i ¼ 1 to S do
if(stopping criterion of si is not met) then
p argmaxp2f1;2;...S;p6¼ig ri;p;
ifrand(0, 1) < ri,p then
Oi Select Nri,p individuals from Op to replace the
individuals in Oi randomly;
endif
fOi Evaluate Oi on transformation si;
Pi Select top N individuals from Pi[Oi based on
fitness value fPi[fOi;
endif;
17. endfor;
18. endwhile;
19. A Select output solution from P by (9).
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