IEEE Computational Intelligence Magazine - February 2021 - 47

where C Dw (M, C L) is the drag coefficient in
Additionally, evaluating airfoil performance in each
the scenario (M, C L) and calculated via an
expensive CFD simulation. Additionally,
scenario requires a CFD simulation that takes minutes.
the ranges of M and C L are defined as
Y = [0.704, 0.734] and C = [0.704, 0.824] in
proposed algorithm takes the advantage of the implicit parallelEquation (9). The geometry of the airfoil is defined by a camism from MFEA and from surrogate models to accelerate the
ber and fixed thickness distribution. Using six Henne-Hicks
search of the minimax EA. For the minimax optimization
functions [60] in Equations (10-15), a camber shape of a
problems in particular, the model management strategy is redesign can be calculated as Equation (16),
designed using a statistical hypothesis test (t-test) to assess the
x 0.5 (1 - x)
optimization progress. According to the experimental results on

(10)
f1 =
e 15x
both benchmark problems and airfoil design problems, the prof2 = sin (rx 0.25) 3 (11)

posed algorithm adaptively updates the surrogate model and
corrects the search direction. The experimental results on both
f3 = sin (rx 0.757) 3 (12)

1.357 3
low- and medium-dimensional problems indicated that the
f4 = sin (rx ) (13)

proposed algorithm can find satisfactory solutions using a very
x 0.5 (1 - x)
limited computational budget. It is worth noting that the pro
(14)
f5 =
e 10x
posed algorithm cannot be applied to the problems with a disf6 = sin (rx 4)(15)

crete scenario space because the joint decision and scenario
6
space becomes mixed that cannot be handled by the employed
y c = y bc + / a i fi, (16)

surrogate model. To solve the minimax problems with a disi=1
crete scenario space, a surrogate model for mixed variables
needs to replace the RBF network with Gaussian radial basis
where y bc is the camber shape of baseline airfoil and a i are
functions in the proposed algorithm.
weights of those six Henne-Hicks functions. Thus, six decision
Although the proposed algorithm inherits the implicit parvariables (a 1, f, a 6) determine the camber shapes and the airallelism of MFEA to efficiently search for optimal solutions for
foil shapes of candidate designs.
worst-case scenarios, it does not use the selective imitation
We compare MMEA, SA-MMEA, MMDE, SA-MMDE,
because of the changing tasks over generations. The tasks in the
MM-MFEA, and SA-MM-MFEA using 250 CFD simulations
population changes over generations, but there could be a part
on the airfoil minimax optimization problem. All compared
of promising tasks that can be kept for a number of generaalgorithms are run for 30 times independently. For each
tions. A new selective imitation should be able to assign both
obtained solution, CFD simulations in 31 × 13 uniform scenew and old tasks to individuals in the population to further
narios are calculated in order to estimate its worst-case perforaccelerate the algorithm. In addition, the worst-case scenario
mance. The results are shown in Table IX. Consistent with the
search for a number of solutions can act as source problems,
results on the benchmark problems, the proposed algorithm
then a multi-source selective transfer framework [61] can be
SA-MM-MFEA outperforms other compared algorithms. In
employed to find the worst-case scenario of other solutions via
other words, with the assistance of multitasking optimization
transfer learning rather than repeated similar search processes.
and surrogate techniques, SA-MM-MFEA can efficiently find
an airfoil with low drag over lift ratio in multiple scenarios.
Acknowledgments
VII. Conclusions

In this work, we have proposed a surrogate-assisted multifactorial evolutionary algorithm to address the challenges of expensive minimax optimization problems for robust design. The

This work was supported in part by the National Natural Science Foundation of China (No. 61976165, 61876025 and
61590922).
References

TABLE IX Optimal solutions obtained by MMEA, SA-MMEA,
MMDE, SA-MMDE, MM-MFEA, and SA-MM-MFEA on the
RAE2822 airfoil problem.
ALGORITHM

OBTAINED SOLUTION

MMEA

0.0098 ± 0.0105

SA-MMEA

0.0123 ± 0.0137

MMDE

0.0055 ± 0.0050

SA-MMDE

0.0070 ± 0.0033

MM-MFEA

0.0077 ± 0.0126

SA-MM-MFEA

0.0031 ± 0.0024

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FEBRUARY 2021 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE

IEEE Computational Intelligence Magazine - February 2021

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