IEEE Computational Intelligence Magazine - February 2021 - 39

f (sw2N, x2N)
sw2N
xN

FIGURE 4 Flow chart of the selection operations in the scenario and decision space, where the candidate scenarios are selected based on their scalar fitness and the candidate solutions are selected based on their estimated worst-case scenario performance.

φ2N
f (s2N, x2N)
f (s2N, x2)
f (s2N, x1)

f (s2, x2N)
f (s2, x2)
Find the
Worst
Scenario
for Each
Solution

=

where w j is the set of the objective values of x j in all the candidate scenarios and R d (s i, w j) is the rank function of s i in w j
in a descending order. Using Equation (6), the scalar fitness { i
represents the worst performance of s i on those P tasks in a

f (sw2, x2)
sw2
Selection
in
Decision
Space

i

f (s2, x1)

f (s1, x2N)
f (s1, x2)
f (s1, x1)
f (sw1, x1)

1
,1 # j # P ,
min R d ^s i, w j h
j
j 1
j
P
w = " f ^ x , s h, f, f ^ x , s h,, (6)
{

Selected
Solutions

	

x2

where the j-th task is to find the worst-case scenario for the
j-th candidate decision x j in the population P. To calculate the
scalar fitness { of each candidate scenario, each scenario-solution pair (x j, s i) in P is evaluated according to its performance
f (x j, s i) as shown in Fig. 4. The scalar fitness { i of s i can be
calculated as below:

sw1

s!r

x1

= argmax f (x j, s), (5)

1 # j #|P|

Estimated
Worst Worst Scenario
Scenarios
Fitness

sj

	

Evaluated Scenario-Solution Pairs

B. Separate Selection Operations
in Scenario and Decision Space

Due to the hierarchical structure of minimax optimization
problems, the selection operations in the scenario and decision spaces of SA-MM-MFEA are hierarchical. For the selection operation in the scenario space, the worst-case scenarios
for the candidate solutions x i are found. Unlike the existing
minimax EAs that serially search the worst-case scenarios of
the candidate x in the population, SA-MM-MFEA formulates such search process as a multitasking optimization
problem for a parallel search. Then, based on those found
worst-case scenarios, the worst-case scenario performance of
each x i can be estimated for the selection operation in the
decision space. Both selection operations are shown in Fig. 4,
where N candidate solutions and scenarios are selected from a
population with 2N individuals.
The selection operation in the scenario space is based on
a scalar fitness function borrowed from MFEA since the
search of the worst-case scenarios of all P solutions is parallelized by formulating as the following multitasking optimization problem:

sN

s2
φ2
Calculate
Scalar
Fitness

φ1

Selection
in
Scenario
Space

s1

Selected
Scenarios
Scalar
Fitness

All the scenario-solution pairs (s, x) in the population are
estimated for the selection operations of scenario and decision
variables. The selection operation in the scenario space is based
on the multifactorial evaluations from MFEA since the worstcase scenario search for all the solutions in the population is
viewed as a multitasking optimization problem. Then, the
selection operation in the decision space is based on the estimated worst-case scenario performance. However, evaluating
scenario-solution pairs are too expensive to conduct the
search. Therefore, we employ the surrogate assistance using an
RBF network that is built in X # r and a generation-based
model management strategy for minimax optimization problems to adaptively update the RBF network. The selection
operations and surrogate assistance, two main contributions of
this work (highlighted in Fig. 3), are explained in the following two sub-sections.

FEBRUARY 2021 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE

39
IEEE Computational Intelligence Magazine - February 2021

Table of Contents for the Digital Edition of IEEE Computational Intelligence Magazine - February 2021
