IEEE Computational Intelligence Magazine - November 2023 - 34
FIGURE 3 The framework of LGSEA.
5 iterations in Algorithm 1) in line 3. A set of trial vectors u,
i.e., the external archive A, is obtained by mapping regular
grid points v from the latent space to the decision space by the
built GTMmodel defined by (5) in line 4.
❏ Guiding Evolutionary Search for Offspring Generation: In lines
7-10, for a randomly selected solution x, a mating pool M
is established with some neighborhood solutions in accordance
with the Euclidean distance in decision space, and a
closest sampled trial vector is selected from the archive A,
as the reference (or guiding) vector, to mate the parents for
offspring generation.
The generated offspring solution updates the population in
line 11. Most existing environment selection methods, mentioned
in Section I, can be utilized in LGSEA.
B. Offspring Generation With LGS
As discussed in Section II-C3, although the regularity is
learned in both MEA/GTM and RM-MEDA, their offspring
generations exhibit low search efficiency in the search space
without the search directions. Therefore, this section introduces
LGS, which aims to enhance search efficiency by guiding
the search with learned manifold information, i.e., some regular
grid points mapped from the latent space to decision space
with the approximation ofmanifolds as the guiding solutions.
Given a solution x randomly selected from the current
population P, the generation procedure y ¼ GenSolðx; u; MÞ
in Algorithm 2 is adopted to generate a new trail solution y by
34 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2023
using the guiding position u and its mating pool M. Inspired
by some existing offspring generation operators, i.e., the operation
ofsimulated binary crossover (SBX) and differential evolution
(DE), three alternative generation operators can be
obtained by LGS in the following ways:
❏ Operator 1:
y ¼
0:5ðð1 þ rrrÞ x þð1 rrrÞ uÞ if rand < 0:5
0:5ðð1 rrrÞ x þð1 þ rrrÞ uÞ else
(9)
❏ Operator 2:
y ¼ u þ rrr ðxr1 xr2Þ
❏ Operator 3:
y ¼ x þ rrr ðu xÞþð1 rrrÞ ðxr1 xr2Þ
(10)
(11)
where Operator 1 is simplified and modified with the SBX
operator, Operator 2 and Operator 3 are the variations of
DE/best/1 and DE/current-to-best/1 mutations, respectively.
xr1 and xr2 are two randomly selected solutions from the mating
pool. To reduce the influence ofparameters on these operators,
all parameters in Operator 1, 2, and 3 are set with
random values, e.g., rand returns a uniformly randomly distributed
value in ½0; 1, rrr is a vector with random values ranging
from 0 to 1.
IEEE Computational Intelligence Magazine - November 2023
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