IEEE Computational Intelligence Magazine - February 2021 - 45

increases, therefore the results in Table IV are
A robust airfoil should be well-performing in a
different from that in Table II. Due to the
large decision space of the test problems, nonvariety of scenarios (i.e., different velocities and
surrogate-assisted EAs (MMEA, MMDE, and
lifts) [5], [59], making the design process a minimax
MM-MFEA) cannot outperform surrogateoptimization process.
assisted EAs (SA-MMEA, SA-MMDE, and
SA-MM-MFEA). In particular, MMDE,
which is the second best algorithm for lowperformance than other surrogate-assisted minimax EAs.
dimensional problems, performs the second worst on those
From the results in both subsections, we can conclude that
median-scale problems.
the proposed algorithm can find the better solution than
Type I problems have separable scenario and decision variexisting methods.
ables, so they are symmetrical problems. Although these problems are the easiest problems among the test problems in this
sub-section, non-surrogate-assisted EAs (MMEA, MMDE, and
C. Experiment on Model Management Strategy
MM-MFEA) cannot outperform surrogate-assisted EAs (SAThe model management strategy in SA-MM-MFEA is differMMEA, SA-MMDE, and SA-MM-MFEA). MM-MFEA is
ent from the existing strategies, because it is designed for
the best performing non-surrogate-assisted EA on type I probexpensive minimax problems. To illustrate the contribution of
lems, and its surrogate version SA-MM-MFEA outperforms all
the statistical hypothesis test-based model management, we
the compared algorithms except for I-Rosenbrock. MMDE is
exclude the statistical hypothesis test-based control for RBF
better than MMEA, but it does not result in the better perfornetwork and promising region size from the proposed model
mance of SA-MMDE than SA-MMEA. The reason has been
management strategy (i.e., LHS is adopted for new samples,
explained in the pervious section: MMDE is more sensitive to
but the sampling region size and the number of hidden layer
approximation errors than MMEA.
nodes of the RBF network are fixed) and compare it with the
Type II problems have both separable and nonseparable
original strategy on the type V problems. We run SA-MMscenario and decision variables, which makes them asymmetriMFEAs with and without statistical hypothesis test-based concal. SA-MM-MFEA has greater advantage than other algotrol for 30 independent times, and the results are presented in
rithms on type II problems than type I problems, where it
Table V and analyzed by the Wilcoxon signed-rank test (sigperforms the best on all the five type II problems. Since type II
nificance level = 0.05) [58].
problems are harder than type I problems, SA-MMDE degenThe results of SA-MM-MFEA(SHT) are significantly better
erates its performance.
than those of SA-MM-MFEA(NSHT) on all the type V probType III problems have a larger number of nonseparable
lems, which indicates the effectiveness of the proposed model
scenario variables than type II problems, and the type IV
management strategy for minimax optimization problems. In
problems have a larger number of nonseparable decision varithe statistical hypothesis test-based model management strategy,
ables than type II problems. The results of compared algothe new data are sampled from an adaptive sampling region
rithms are similar to the results on type II problems, where
around the predicted best solution, and the number of hidden
SA-MM-MFEA has the best performance and MMEA has
layer nodes of the RBF network changed over the optimizathe worst performance. The only difference is that SAtion process, which enables SA-MM-MFEA to further update
MMEA obtained the best results on III-Rosenbrock. Type V
the RBF network in a small local area and to exploit the
problems have nonseparable scenario and decision variables,
promising region.
which is the hardest type among those five types. SA-MMMFEA still outperforms other compared algorithms on all
five type V problems.
TABLE V MSE of SA-MM-MFEAs with and without the
According to the Friedman ranks in Table IV, the perforstatistical hypothesis test-based control (short for SA-MMmance of compared algorithms changes slightly for different
MFEA(SHT) and SA-MM-MFEA(NSHT)) on the type V
types of problems. When the computational budget is limited
problems. The results are shown in the form of mean ±
standard deviation. The results are analyzed by the
for solving medium-scale minimax optimization problems,
Wilcoxon signed-rank test (significance level = 0.05). The
the main challenge for minimax EAs is the large search space
significant algorithms for each problem are highlighted in
rather than their asymmetry. While general expensive probbold face.
lems with 30 decision and scenario variables are still challengSA-MM-MFEA(SHT)
SA-MM-MFEA(NSHT)
ing for EAs, expensive minimax optimization problems with
V-GRIEWANK
0.0 ± 0.0
0.5 ± 0.1
30 decision variables are even harder. With the help of surroV-ACKLEY
0.0 ± 0.1
0.3 ± 0.2
gate model, the proposed algorithm significantly reduces the
V-RASTRIGIN
0.0 ± 0.0
0.7 ± 0.2
computation cost, enabling it to outperform MMDE. Furthermore, the MTO technique employed in the proposed
V-ELLIPSOID
0.0 ± 0.0
0.6 ± 0.2
algorithm enhances the search efficiency and leads the better
V-ROSENBROCK
4.4 ± 2.4
6.6 ± 2.7

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IEEE Computational Intelligence Magazine - February 2021

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