This document summarizes a research paper that proposes using a Bak-Sneppen model to self-adaptively control three parameters (ω, c, ρ) of particle swarm optimization (PSO). The BS-PSO method was tested on benchmark optimization problems and showed competitive performance compared to other PSO variants without requiring hand-tuning of parameters. Future work is suggested to test the method in dynamic environments, investigate scalability, and further study the critical state behavior of the Bak-Sneppen model.

1. Carlos M. Fernandes1,2
J.J. Merelo1
Agostinho C. Rosa2
1Departmentof computers architecture and technology, University of
Granada, Spain
2 LaSEEB-ISR-IST, Technical Univ. of Lisbon (IST), Portugal

2. Why?
Exploration
Exploitation
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3. SOC
Deterministic: parameter
values change according to Adaptive: variation depends
deterministic rules indirectly on the problem and
search stage
Self- adaptive: values
evolve together with the
solutions to the problem Hand-tuning
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4. Bio-inspired: bird
flock and fish
school.
Cultural and social
interaction:
cognitive, social and
random factors.
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5. X i – position of particle i (vector)
V i – velocity of particle i (vector)
X i(t) =Xρ Xi(t-1)+Xi(t-1)+Vi(t)
i(t-1)+Vi(t)
Vi(t) = Vω Vi(t-1)+c1 r1(pi-xi(t-1))+c2 r2(pg-xi(t-1))
i(t-1)+c1 r1(pi-xi(t-1))+c2 r2(pg-xi(t-1))
c
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6. SOC: a state of criticality
formed by self-organization
in a long transient period at
the border of order and
chaos.
The Sandpile Model
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7. Power-laws
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8. [1] BK-inspired Extremal Optimization, Boettcher and
Percus, 2003
[2] Sandpile in Evolutionary Algorithms, Krink et al., 2000-
2001
[3] SOC in PSO, Løvbjerg and Krink, 2004
[4] BK model in Evolutionary Algorithms – Self-Organized
Random Immigrants GA (SORIGA), Tinós and Yang, 2008
[5] Sand Pile Mutation for Genetic Algorithms, Fernandes et
al., 2008-2012
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9. Per Bak (How Nature Works):
Random numbers are arranged in
a circle. At each time step, the f = 0.41
0.16 f = 0.55
lowest number, and the numbers
at its two neighbours, are
replaced by new random f f= 0.32
= 0.14
numbers.
f = 0.91
f = 0.79
f = 0.23
f = 0.90
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11. X i(t) = ρ Xi(t-1)+Xi(t-1)+Vi(t)
Vi(t) = ω Vi(t-1)+c1r1(pi-xi(t-1))+c2r2(pg-xi(t-1))
ω = 1-bs_fitness(i)
c1 =c2=1+bs_fitness(i)
ρ =random [0, 1-bs_fitness(i)]
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12. ω
BS model bs_fitness c PSO
ρ
BS species
Particles
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13. o Sphere, Rastrigin, Rosenbrock, Griewank
o lbest and gbest topologies.
o TVIW-PSO, RANDIW- PSO, GLbestIW-PSO and IA-PSO
o Population size: n = 20
o 3000 generations
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14. (ω, c, ρ)
bs, 1.49, 0 bs, 2.0, 0 bs, bs, 0 bs, bs, 0.25 bs. bs, bs
f1 3.35e+01 1.38e-15 8.30e-32 0.00e+00 0.00e+00
(1.90e+02) (3.21e-15) (3.47e-31) (0.00e+00) (0.00e+00)
f2 1.67e+05 1.88e+02 8.56e+01 2.61e+01 2.60e+01
(1.17e+06) (2.53e+02) (7.98e+01) (2.66e-01) (1.58e-01)
f3 2.82e+02 1.11e+02 2.02e+02 4.88e+00 3.32e+00
(4.44e+01) (2.75e+01) (4.16e+01) (7.73e+00) (7.09e+00)
f4 1.63e+00 1.25e-02 1.65e-02 3.79e-03 4.51e-03
(5.93e+00) (1.26e-02) (2.24e-02) (2.29e-03) (4.00e-03)
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15. full control
PSO 1 vs. PSO 2 f1 f2 f3 f4
BS-PSO vs TVIW-PSO + + + +
BS-PSO vs RANDIW-PSO + + + +
BS-PSOvs GLbestIW-PSO + + + +
without perturbation of position
PSO 1 vs. PSO 2 f1 f2 f3 f4
BS-PSO vs TVIW-PSO + + – –
BS-PSO vs RANDIW-PSO + ~ – ~
BS-PSO vs GLbestIW-PSO + + ~ +
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16. PSO 1 vs. PSO 2 f1 f2 f3 f4
BS-PSO vs IA-PSO + + ~ +
BS-PSOvs IA-PSO (bs controled ) ~ ~ + ~
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17. PSO 1 vs. PSO 2 f1 f2 f3 f4
BS-PSO vs TVIW-PSO + + + +
BS-PSO vs RANDIW-PSO + + + +
BS-PSOvs GLbestIW-PSO + + + +
BS-PSO) vs IA-PSO + + ~ +
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18. 1
0.8
0.6
0.4
0.2
0
0 100 200 300 400 500 600 700 800 900 1000
iterations
1000
number of samples
100
10
1
0.01 0.1 1
ωi =ρi
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19. o With a simple set of equations we are able to
control three (four) parameters of the PSO.
o The resulting algorithm is competitive with other
variants of the PSO.
o Full control of the PSO by BS attains good
performance.
o Hand-tuning is not required.
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20. o Information (state) from PSO into the model.
o Test BS-PSO on dynamic environments.
o Scalability.
o BS critical state: investigate the behaviour
before and after the system reaches the critical
state.
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