IEEE Computational Intelligence Magazine - February 2021 - 38

As mentioned in Section II, the difficulty of
minimax optimization problems is the prohibitive computation costs spent on infinite maximization subproblems. The optimization
processes of those similar maximization subproblems in a population share some mutual
knowledge, which can be formulated as a
multitasking optimization problem. Therefore,
we believe that the implicit parallelism of the
population and positive knowledge transfer of
MFEA using the multifactorial scalar fitness can improve the
effectiveness and efficiency of the evolutionary search on minimax optimization problems.

In this work, we borrow the implicit parallelism of the
multifactorial scalar fitness in MFEA and the cheap
cost of the RBF network to the proposed minimax
optimization algorithm (SA-MM-MFEA) to address the
issue of a limited computational budget for expensive
minimax optimization problems.
its scalar fitness { is assigned as 1/rx. Then, during the optimization process, each solution will only be evaluated on the
task of its skill factor.
The main steps of MFEA are similar to conventional EAs,
but the details of variation and evaluation are changed for multitasking as below:
❏❏ Initialization: MFEA generates an initial population P,
evaluates P on all the tasks, and calculates their skill factors
x and scalar fitness {.
❏❏ Stopping condition: If the stopping condition is satisfied,
MFEA outputs the obtained optimum of k tasks; otherwise,
it remains in the loop of variation, evaluation and selection.
❏❏ Variation: MFEA generates an offspring population O via
assortative mating, where crossover is applied to those two
parents if two parents have a same skill factor or it is at a
pre-defined probability; otherwise, mutation is applied to
those two parents.
❏❏ Evaluation: Calculating one scalar fitness { requires k
function evaluations. When k is a large number, the fitness
calculation would be computationally demanding. Therefore, MFEA assigns O skill factors via selective imitation,
which depends on the skill factors of their parents. Then,
only the assigned tasks of O are evaluated, while the objective values of other unevaluated tasks are set as a very large
number.
❏❏ Selection: MFEA re-calculates the scalar fitness of P , O
and selects half individuals with the best scalar fitness as the
parent population for the next generation. Then, MFEA
goes to the step " Stopping condition. "

IV. Proposed Algorithm

SA-MM-MFEA follows the process of generic SAEAs, which
includes model building, evolutionary operations (population
initialization, variation, and selection), and model management
[30]. The diagram of the proposed algorithm is illustrated in
Fig. 3. At the very beginning of SA-MM-MFEA for solving a
minimax optimization problem in Equation (2), a number of
training data in the joint decision and scenario space (X # r)
are sampled using Latin hypercube sampling (LHS) to build an
initial RBF network. Then, a population P with N random
scenarios and solutions is initialized, where an individual is
encoded as (s, x). Before the computational budget (a fixed
number of expensive function evaluations) is exhausted, the
proposed algorithm repeats the evolutionary search loop of
variation (simulated binary crossover and polynomial mutation), evaluation, selection, and model management, where the
evaluation, selection, and model management are different from
the existing SAEAs.

Model Management
Strategy in Every Ng
Generations
Build Initial
RBF Network

Population
Initialization

Stop?

Variation

Evaluation
Scenario-Solution
Pairs Using RBF Network
Surrogate Assistance

Output Optimal
Solution

FIGURE 3 A diagram of surrogate-assisted minimax multifactorial evolutionary algorithm.

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | FEBRUARY 2021

Multifactorial-Based
Selection in Scenario
Space
Worst PerformanceBased Selection in
Decision Space
Selection

IEEE Computational Intelligence Magazine - February 2021

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