IEEE Computational Intelligence Magazine - February 2021 - 41

the population has concentrated in a wrong area after N g
generations, which needs a re-initialization to correct the
search direction.
❏❏ Case 2 (RBF is performing well): The average objective
value of D N (A N ) is better than that of D T (A T ) and PV is
larger than 0.05, which shows the right search direction of
the last N g generations but the objective value is not significantly improved. The proposed algorithm keeps the settings
of the RBF network and the population for the next N g
generations.
❏❏ Case 3 (RBF is performing well and a promising
region is found): D N is sampled in a promising area
which is predicted with best worst-scenario objective values, so it is very rare that D N is significantly better than D T
which is randomly sampled in the whole solution-scenario
joint space, i.e., the average objective value of D N (A N ) is
better than that of D T (A T ) and PV is smaller than 0.05. In
that case, the proposed algorithm has found a very promising region. To further exploit this promising local area, the
proposed algorithm re-trains the RBF network with one
hidden node using D N .
The initial 11 (d + c) training data are sampled using LHS,
i.e., each dimension in the feasible space X # r is divided into
10 grids, so we set the initial l as 10% of the range of each
dimension. When the model management strategy meets progressing cases (2 and 3), the population will further exploit the
promising local area by reducing l to 0.5l . After the re-setting
for the RBF network, population, and sampling region size, the
RBF network is retrained using all the training data.
V. Empirical Study

In this section, we compare SA-MM-MFEA with the state-ofthe-art minimax EAs on both low- and medium-dimensional
benchmark problems. The following algorithms are compared:
❏❏ MMEA (minimax evolutionary algorithm) is an EA with
the worst-case scenario performance as its fitness function.
The worst-case scenario for each candidate solution is
obtained via extra evolutionary search. Since MMEA does
not use any mechanism for saving evaluations, it is the baseline algorithm.
❏❏ SA-MMEA (surrogate-assisted minimax evolutionary algorithm) replaces the real function evaluation with the RBF
network in MMEA. In each generation, as most SAEAs
[33], two solutions in the population are re-evaluated to
enrich the training data: the best solution and the most different solution compared with the training data.
❏❏ MMDE [19] is a differential evolution algorithm for minimax optimization problems, where a bottom-boosting
scheme is employed to detect promising solutions and a
partial-regeneration strategy is used to enhance the exploration ability.
❏❏ SA-MMDE (surrogate-assisted MMDE) replaces the real
function evaluation with the RBF network. In each generation, two solutions in the population are re-evaluated to
enrich the training data: the best solution and the most dif-

ferent solution compared with the training data. The comparison between MMDE and SA-MMDE can show the
effect of surrogate models on MMDE.
❏❏ MM-MFEA (minimax multifactorial evolutionary algorithm) is an SA-MM-MFEA variant without surrogate

Joint Scenario and
Decision Space
Determine
Sampling
Region

LHS

Statistical Hypothesis Testing (DT Versus DN)
Case 2:
AN < AT or
PV > 0.05

Case 1:
AN > AT or
PV > PVbest
f

Case 3:
AN < AT or
PV < 0.05
f

AN
DT

Add DN to
Training Data

Set DN as
Training Data

Increase
Complexity
of RBF
Network by
Adding One
Hidden Node

Keep
Complexity
of RBF
Network
Unchanged

Reduce
Complexity
of RBF
Network by
Using One
Hidden Node

Re-Build RBF Network Using Updated
Training Data

Re-Initialize
Population

Reduce
Sampling
Region

DT: Initial Training Data
Predicted Best Solution in the Last
Model Management
Current Predicted Best Solution
DN : New Samples
Sampling Region
FIGURE 5 Three cases in the new generation-based model management strategy.

FEBRUARY 2021 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE

IEEE Computational Intelligence Magazine - February 2021

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