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# When Sharing Fails

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## Abstract and Figures

Sharing, introduced by Goldberg and Richardson in 1987, is probably one of the most investigated ideas for multimodal optimization. Empirical tests have indicated that sharing is capable of maintaining multiple peaks located simultaneously -- a feature that allows a final human selection among the found solutions. In this paper I present a theoretical argument regarding the performance of sharing. The argument is supported with a series of tests on variants of a simple problem, which is one of Goldberg and Richardson's original test function where a constant is added. The results from these tests indicated that sharing is very sensitive to the range of fitness values. Finally, three extensions of sharing are proposed and discussed.
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σ
share
F it
0
(I
i
) =
F it(I
i
)
P
µ
j=1
sh(d(I
i
, I
j
))
F it
0
(I
i
) F it(I
i
)
µ sh(d)
d(I
i
, I
j
)
I
i
I
j
sh(d) =
(
1 (d/σ
share
)
α
d < σ
share
0
σ
share
α
α
σ
share
x [0, 1]
F
1
(x) = sin
6
(5.1πx + 0.5)
F
2
(x) = sin
6
(5.1πx + 0.5) · e
(4 ln(2)
(x0.1)
2
0.8
2
)
F
1
(x)
F
1
(x)
F
2
(x)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
F
1
(x) = sin
6
(5.1πx + 0.5)
F
1
(x)
F
3
(x) = F
1
(x) + C
sh(d) =
(
1 d < σ
share
0
C = 10
F
0
3
(x) = F
1
(x) + 10 F
0
3
(x)
F it
0
(I)
11
1 + 1
= 5.5
σ
share
F
0
3
(x) = F
1
(x) + 10
σ
share
σ
share
= 0.1 α = 1 C = 10
sh(d)
0.9 · σ
share
popsize = 50
p
c
= 0.8
#bits = 30
#iterations = 100
F
3
(x) = F
1
(x) + C
C
C
p
m
= 0.0 p
m
=
0.03333
C = 0
p
m
= 0.0 p
m
= 0.03333
popsize = 50
C = 0 C = 1
C = 2 C = 5 C = 10
C = 0
C =
1
C = 2 C = 5
C = 10
F it
0
(I
i
) =
exp(σ
trans
· F it(I
i
))
P
µ
j=1
sh(d(I
i
, I
j
))
σ
trans
σ
trans
p
m
p =
capacity
σ
capacity
F
1
(x)
σ
capacity
= 0.1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
p
m
= 0.0
p
m
= 0.03333 C = 0 C = 1 C = 2
C = 5 C = 10 y x
popsize = 50
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1
p
m
= 0.0 p
m
= 0.03333
C = 0 C = 1 C = 2 C = 5
C = 10 y x
popsize = 50
... ? Select different niche mechanisms as mentioned in [10]. ...
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Conference Paper
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Article
Thesis (Ph. D.)--University of Michigan, 1975. Includes bibliographical references (leaves 253-256). Photocopy.