IEEE Computational Intelligence Magazine - February 2023 - 88

In Algorithm 1, the involved functions,
including CollectðÞ, RepresentðÞ
and the Q-table or network, depend on
specific EA. In the following sections,
this framework is applied to two well
known hybrid EAs, i.e., LSHADE [12]
andHSES [17].
IV. APPLYING THE FRAMEWORK
TO LSHADE
A. Hybrid Population Size Strategy
The population size is an essential
parameter for EA, which is often fixed
during the search procedure in most
EAs. LSHADE [12] applies a simple
scheme to control the population size,
called linear population size reduction
(LPSR) strategy. The scheme takes the
population size as a linear function of
the number offitness evaluations (nfes)
with negative slope. The LPSR strategy
can be applied as follows:
Ngþ1 ¼
nfes þNini
NminNini
maxNFEs
(1)
where Nmin;Nini
are pre-fixed constants
representing the minimum population
size and initial population size,
respectively; and ½a rounds a to its nearest
integer. Some DE variants, such as
iL-SHADE [26] and jSO [27], also
apply the LPSR.
In the algorithms employing LPSR,
LPSR is applied from the very beginning
of the evolutionary search. However,
it might be more appropriate to
apply it whenever necessary. In light of
this idea, our goal is to control when to
apply the LPSR strategy. Thus the
Algorithm 1. The Algorithmic Framework based on Adaptive Structural Parameter Control
Input: an optimization function fðxÞ; x 2 R
d~
, an initial population X0 2 R
maximum number of
fitness evaluation maxNFEs, a Q network
Qð;; wÞ and horizon T.
Output: an optimal solution x
1: Initialization: X0, G0, F ; and X ;;
2: Set G EA1ðÞ; Q NFEs;
3: for t ¼ 1 ! T do
4:
5: F F Ft;
6: X X Xt;
½Xt; Ft; Gt GðXt1; f; Q; Gt1Þ;
S
S
7: Ut CollectðX; FÞ;
8: st RepresentðUtÞ;
9: at arg maxa2AfQðst; a; wÞg;
10: if at ¼¼ 1or t ¼¼ T then
11: G EA2ðÞ;
12:
Exit;
13: else
14: G EA1ðÞ;
15: end
16: end
17: ½X; F; Gtþ1 GðXt; f; maxNFEs tQ; GtÞ;
18: return x arg min F.
update formula of the population size is
proposed as follows:
Ngþ1=
degenerates to LPSR (resp. the prefixed
population size). Different nfess
could result in significantly different
(hi
NminNini
maxNFEsnfessðnfes-nfessÞ+Nini
Ng
where nfess is the pre-defined parameter
used to decide when to use LPSR.
It is seen that when nfes < nfess,
the population size is fixed, while
LPSR is used otherwise. Figure 2 shows
how different nfess
values affect the
population size. It is seen that when
nfess ¼ 0 (resp. nfess ¼ maxNFEs), it
if nfesnfess;
otherwise,
(2)
performance. The proposed population
size updating strategy is named as hybrid
population size strategy (HPSS).
B. Q-LSHADE
This section presents how to apply the
proposed framework (Algorithm 1) to
LSHADE with HPSS. Note that
LSHADE equipped with or without
LPSR are regarded as two EAs.
For clarification, LSHADE with fixed
population size is denoted as LSHADEf,
and the original LSHADE is denoted as
LSHADEo. Then the proposed framework
is used to control when to switch
from LSHADEf to LSHADEo.
The evolution procedure ofLSHADE
is summarized in SupplementaryMaterials.It
canbewritten in aconcise formas follows:
½Xt; f~fk
FIGURE 2 The effect of nfess on the population size reduction.
88 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | FEBRUARY 2023
bestgGmax
k¼0 ; Gt
¼ LSHADEoðXt1;f; NFEs; Gt1Þ;
d~N,
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