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The Sandpile Mutation Operator for Genetic

Algorithms

Carlos M. Fernandes1,2

J.L.J. Laredo1

J.J. Merelo1

Agostinho C. Rosa2

1Department of Architecture and Computer Technology, University of Granada, Spain

2 LaSEEB-ISR-IST, Technical Univ. of Lisbon (IST), Portugal

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Summary1. Motivation: Dynamic Optimization

2. Self-Organized Criticality and the Sandpile

3. Sandpile Mutation GA (GGASM)

4. Results

5. Mutation Rates

6. Conclusions

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Dynamic Optimization Industrial applications (non-linearities,

multi-objective) Some applications have dynamic

components• Fitness function depends on time t• Non-stationary (or dynamic) optimization

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Dynamic Optimization

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• Classification of dynamic problems• severity• period between changes• others...

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Dynamic Optimization and EC

Reaction to Changes

Memory Schemes

Multi-Population Schemes

Diversity Maintenance• Random Immigrants GA (RIGA)

one additional parameter

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Parameter Control

Deterministic: parameter values change according to deterministic rules

Adaptive: variation depends indirectly on the problem and the search stage

Self- adaptive: the values to evolve together with the solutions to the problem

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Sand Pile Mutation Operator

“Sand” is dropped on top of 2D lattice

When the number of grains exceed the critical value, there is an avalanche of sand

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Self-Organized Criticality (SOC)

SOC is a state of criticality

formed by self-

organization in a long

transient period at the

border of order and chaos.

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SOC in Evolutionary Computation

Krink and co-authors

• the power-law is computed offline

Self-Organized Random Immigrants GA

• uses a SOC model to introduce random

immigrants in the population

Sandpile Mutation

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Sand Pile Mutation Operator

l1

l2

l3

…

0

1

2

3

4

n1

n2

n3

…

Z

0

1

2

3

4Z Drop (g) grains (g is grain

rate) If h(x,y) > 3, topple Maximization: mutates if rand (0,1.0) > (normalized) fitness

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Parents’ fitness

The lattice is the population

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Dynamic Optimization Problems

[Yang & Yao] problem generator• period between changes (ε)• severity (ρ)

ε : 1200, 2400, 24000, 48000 ρ : 0.05, 0.3, 0.6, 0.95

• each stationary problem -> 16 different dynamic problems

fast and severe changes

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Dynamic Optimization Problems

m-k trap functions (30 and 40 bits)

Royal Road (64 bits)

Knapsack (100 bits)

non-deceptive, nearly-deceptive, deceptive,

combinatorial (constrained)

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Performance Metrics

Offline performance

• average of the best fitness throughout the run

Statistical tests

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Algorithms

Generational Standard GA (GGA)

Self-organized Criticality RIGA (SORIGA)

Elitism-based Immigrants GA (EIGA)

GGA with Sandpile Mutation (GGASM)

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Tests Several mutation probability and

population size values.

binary tournament 2-elitism uniform crossover (p=1.0)

• Balance disruptive effect and selective pressure

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Sand Pile Mutation

ρ →ε = 1200 ε = 2400 ε = 24000 ε = 48000.05 .3 .6 .95 .05 .3 .6 .95 .05 .3 .6 .95 .05 .3 .6 .95

order-4

GGA − − ≈ ≈ ≈ ≈ ≈ ≈ + + + + + + + +

SORIGA ≈ + + + ≈ + + + + + + + + + + +

EIGA − − ≈ − − ≈ ≈ − + + + + + + + +

R. Road

GGA + ≈ ≈ ≈ ≈ + ≈ ≈ ≈ + + + + + + +

SORIGA + + + ≈ + + + ≈ + + + + + + + +

EIGA ≈ ≈ + ≈ + + + ≈ + + + + + + + +

Knapsack

GGA − − + + − ≈ + + + + ≈ − + + + ≈

SORIGA − ≈ + + ≈ ≈ + + + + ≈ ≈ + + + ≈

EIGA − − + + − − + + + + − − + + + −

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Sand Pile Mutation Rate

3 246 489 732 975 1218146117041947219024332676291931623405364838910

0.1

0.2

0.3

0.4

0.5ρ = 0.05

online m

uta

tion r

ate

3 246 489 732 975 1218146117041947219024332676291931623405364838910

0.1

0.2

0.3

0.4

0.5ρ = 0.6

online m

uta

tion r

ate

ε = 24,000order-4 trap

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Conclusions GGASM is able to improve other GAs’

performance on dynamic optimization

Parameter set is not increased

Sandpile mutation rate self-regulates: mutation rates’ distribution depends on the problem and dynamics

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Future Research The Sandpile mutation may be hybridized

with any kind of Evolutionary Algorithm (and other bio-inspired algorithms)

Study the mutation rates and mutation distribution.

Test the Sandpile Mutation on stationary problems and constrained dynamic optimization

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