IEEE Computational Intelligence Magazine - February 2023 - 66

Appendix
A. Proof of Theorem 1
Proof. The basic idea of truncation selection is that the top a(t)
individuals in the fitness ranking of the current population are
selected as the next generation population, which can be modeled
as,
psðx; t þ 1Þ¼
(
where aðt þ 1Þ¼ ssðxÞbsðtþ1Þ qsðx; tÞ dx is a real number. Since
R
the following formula holds,
qsðx; tÞ¼½1 r2
j ¼ argmax
j2f1;2;...;sg; j 6¼s
we have,
Z
rs;jðtÞ¼
s;j ðtÞ psðx; tÞþ r2
PN PN
m¼1
s;j ðtÞpj
n¼1 Iðxm
N N K
qs x; tðÞ dx
ssðxÞbsðtÞ
Because ssðxÞbsðtÞ psðx; tÞ dx ¼ 1, we have,
Z
R
1 r2
s;jðÞt
qsðx; tÞ dx:
ssðxÞbsðtÞ
According to Assumptions 1 4), the following inequality
holds,
aðt þ 1Þ < 1 r2
Then,
Z
qsðx; tÞ dx <
ssðxÞbsðtþ1Þ
Z
s;jðÞt :
(22)
(21)
Z hi
1 r2
ssðxÞbsðtÞ
s;jðÞt
ðx; tÞ
s ðtÞ;xn
j ðtÞÞ :
(19)
qsðx;tÞ
aðtþ1Þ ;
0;
ifssðxÞ bsðt þ 1Þ
ifssðxÞ < bsðt þ 1Þ
;
(18)
Because ps(x,0) is positive and continuous in search space X
(Assumptions 1 3)), we obtain,
lim
t!1
Let X0 ¼ {x jx2X, ss(x) > ss
obtain,
ps x; tðÞ dx: (20)
lim
Z
t!1 X0
ps x; tðÞ dx ¼þ1:
(27)
Because (27) contradicts the fact that ps(x,t) is a probability
density function, we have,
lim
t!1
Due to E½ssðxÞ ¼ X ssðxÞ psðx; tÞ dx bsðtÞ , we have,
Z
R
lim E ss xðÞ
t!1
½	¼
X
ss xðÞ ps x; tðÞ dx ¼ s
s :
(29)
bsðtÞ¼ s
s :
(28)
ps x; tðÞ ¼ þ1; 8x : ss xðÞ > s0
s:
(26)
0}. According to Fatou's lemma, we
i.e., bsðtÞ < bsðt þ 1Þ, which implies that there exists a limit,
lim
t!1
bsðÞt ¼ s0
Assume that ss
0  s0
s:
(24)
0 ¼ ss
.
(25)
qsðx; tÞ dx;
ssðxÞbsðtÞ
(23)
by the Major Research Plan of the National Natural Science
Foundation of China under Grants 91438201, 91438103, and
91838303; in part by the National Natural Science Foundation
of China under Grants U1701267, 62076192, 62006177,
61902298, 61573267, and 61906150; in part by the 111 Project;
in part by the Program for Cheung Kong Scholars and
Innovative Research Team in University under Grant IRT
15R53; in part by the ST Innovation Project from the Chinese
Ministry of Education; in part by the Key Research and
Development Program in Shaanxi Province of China under
Grant 2019ZDLGY03-06; in part by the National Science
Basic Research Plan in Shaanxi Province of China under
Grants 2019JQ-659 and 2022JQ-607; in part by the Scientific
Research Project of Education Department In Shaanxi Province
of China under Grant 20JY023; in part by the Fundamental
Research Funds for the Central Universities under
Grants XJS201901, XJS201903, JBF201905, and JB211908;
and in part by the CAAI-Huawei MindSpore Open Fund.
66 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | FEBRUARY 2023
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