IEEE Computational Intelligence Magazine - February 2021 - 37

x4

x

x3

x2
x1

Maximization
Subproblem 1

f

f

s

Maximization
Subproblem 4

Maximization
Subproblem 3

Minimization

Maximization
Subproblem 2

f

s

f

s

s

FIGURE 1 Illustration of minimax optimization problems, where the upper level is the minimization problem in the decision space and the lower
level is infinite maximization problems in the scenario space.

which is conflicting for asymmetr ical problems as
f (x ), s) # f (x ), s )) # f (x, s )) is not satisfied. Therefore, minimax coevolutionary algorithms cannot find the global optimum [19]. In addition, minimax optimization problems
become even more challenging for EAs when a single calculation of f (x, s) is expensive.
III. Preliminaries of Multitasking Optimization

Conventional EAs serially solve different problems using
a single population. As discussed in the previous section, a
minimax optimization problem can be decomposed into
an infinite number of maximization subproblems, making
EAs inefficient for those massive subproblems. However,
evolutionary multitasking optimization is an exception as
it can simultaneously solve k tasks (optimization problems
{x 1, f, x k} = argmin { f1 (x), f, fk (x)}) using a unified population [42].
MFEA [42] is a representative evolutionary multitasking
optimization framework. Inspired by multifactorial inheritance [56], MFEA makes use of the diversity in the population to solve multiple problems by involving factorial rank r,
skill factor x, and scalar fitness {. Firstly, MFEA represents
the decision variables of different tasks in a unified population
and evaluates the solutions on all the tasks. Each solution will
then be assigned a skill factor x to label its best performing
task. To be specific, if r j is the rank of a solution in the population on the j-th task, then it is associated with the task
x = argmin (r j) with the best factorial rank among k tasks, and
j

Solution
Population
Px

Scenario
Population
Ps

x

s

Fitness h (x, Ps)

Selection

Fitness g (s, Px)

Selection

FIGURE 2 Parallel optimization structure in minimax coevolutionary
algorithm. The fitness functions of Px and Ps for the solution and scenario space depend on each other.

FEBRUARY 2021 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE

37



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