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A Cooperative Approach to Particle Swarm Optimization Authors : Frans van den Bergh, Andries P. Engelbrecht Journal : Transactions on Evolutionary Computation, vol 8, No. 3, June 2004 Presentation : Jose Manuel Lopez Guede Introduction

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A Cooperative Approach to Particle Swarm Optimization

Authors: Frans van den Bergh, Andries P. Engelbrecht Journal: Transactions on Evolutionary Computation, vol 8, No. 3, June 2004

Presentation: Jose Manuel Lopez Guede

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Introduction

“Curse of dimensionality”
PSO
CPSO
CPSO-Sk
CPSO-Hk
GA comparation
Results

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Particle Swarm Optimizers I

PSO:

– Stochastic optimization technique – Swarm: a population – During each iteration each particle accelerates influenced by:

Its own personal best position
Global best position

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Particle Swarm Optimizers II

– –

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Particle Swarm Optimizers III

– During each iteration, each particle is updated:

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Particle Swarm Optimizers III

– During each iteration, each particle is updated: – The global best position is updated:

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Cooperative Learning I

PSO:

– Each particle represents an n-dim vector that can be used as a potential solution.

position Best position

f the particle

Best position

f the swarm

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Cooperative Learning II

– Drawback:

Authors show a numerical example where PSO goes to a

worst value in an iteration.

Cause: error function is computed only after all the

components of the vector have been updated to their new values.

– Solution:

Evaluate the error function more frecuently.
For every time a component in the vector has been updated.

– New problem:

The evaluation is only possible with a compete vector.

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Cooperative Learning III

CPSO-S:

– n-dim vectors are partitioned into n swarms of 1-D – Each swarm represents 1 dimension of the problem – “Context vector”:

f requires an n-dim vector
To calculate the context vector for the particles of swarm j,

the remainig components are the best values of the remaining swarms.

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Cooperative Learning IV

context vector

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Cooperative Learning V

Advantage:

– The error function f is evaluated after each component in the vector is updated.

However:

– Some components in the vector could be correlated. – These components should be in the same swarm, since the independent changes made by the different swarms have a detrimental effect on correlated variables. – Swarms of 1-D, and swarms of c-D, taken blindly.

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Cooperative Learning VI

CPSO-Sk:

– Swarms of 1-D, and swarms of c-D, taken blindly, hoping that some correlated variables end up in the same swarm. – Split factor: The vector is split in K parts (swarms) – It is a particular CPSO-S case, where n=K.

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Cooperative Learning VII

CPSO-S

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Cooperative Learning VII

– Drawback:

It is possible that the algorithm become trapped in a state

where all the swarms are unable to discover better solutions: stagnation.

Authors show an example.

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Hybrid CPSOs – CPSO-Hk I

Motivation:

– CPSO-Sk can become trapped. – PSO has the hability to scape from pseudominimizers. – CPSO-Sk has faster convergence.

Solution:

– Interleave the two algorithms.

Execute CPSO-Sk for one iteration, followeb by one iteration
f PSO.
Information interchange is a form of cooperation.

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f the swarm Q

PSO Swarms of the CPSO-Sk

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Experimental Setup I

Compare the PSO, CSPO-Sk, CSPO-Hk algorithms.
Measure: #function evaluations.
Several popular functions in the PSO comunity

were selected for testing.

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Experimental Setup II

“all the functions where tested under coordinate rotation using Salomon’s algorithm”

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Experimental Setup III

PSO configuration:

– All experiments were run 50 times – 10, 15, 20 particles per swarm. – Results reported are averages os the best value in the swarm.

Domain: “magnitude to which the initial random particles are scaled”

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Experimental Setup IV

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Experimental Setup V

GA configuration:

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Experimental Setup VI

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Results I

Fixed-Iteration Results I

– 2.10^5 function evaluations.

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Results II

Fixed-Iteration Results II

– 2.10^5 function evaluations.

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Results III

Fixed-Iteration Results III

– PSO-based algs. performed better that GA algs. in general. – Cooperative algorithms collectivelly performed better than the standard PSO in 80% of the cases.

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Results IV

Robustness and speed Results I

– “Robustness”: the algorithm succeed in reducing the the f below a specified threshold using fewer that than a number of evaluations. – “A robust algorithm”: one that manages to reach the threshold consistentle (during all runs).

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Results V

Robustness and speed Results II

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Results VI

Robustness and speed Results III

– CPSO-H6 appears to be the winner because it achieved a perfect score in 7 of 10 cases. – There is a tradeoff between the convergence speed and the robustness of the algorithm.