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Particle Swarm Optimization Algorithms to Continuous Problem. Monday, March 10, 2014 by. Yoon-Teck Bau, Hong-Tat Ewe, Chin-Kuan Ho Faculty of Information Technology Multimedia University, Malaysia {ytbau, htewe, ckho}@mmu.edu.my http://pesona.mmu.edu.my/~ytbau/. Talk Outlines.

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Particle swarm optimization algorithms to continuous problem l.jpg

Particle Swarm Optimization Algorithms to Continuous Problem

Monday, March 10, 2014

by

Yoon-Teck Bau, Hong-Tat Ewe, Chin-Kuan Ho

Faculty of Information Technology

Multimedia University, Malaysia

{ytbau, htewe, ckho}@mmu.edu.my

http://pesona.mmu.edu.my/~ytbau/


Talk outlines l.jpg
Talk Outlines

  • Research Objective

  • Particle Swarm Optimization (PSO) Algorithms Overview

  • PSO to Continuous Problem

  • PSO and Non-linear Maximization Problem

  • Experiments and Results

  • Conclusions

  • References


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Research Objective

  • To study PSO in continuous problem

  • To compare the performance of genetic algorithms with PSO in maximization problem

  • To share and exchange knowledges related to PSO and swarm intelligence


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PSO Algorithms Overview

  • Introduced by Russel Ebenhart (an Electrical Engineer) and James Kennedy (a Social Psychologist) in 1995

  • Belongs to the categories of Swarm Intelligence techniques and Evolutionary Algorithms for optimization

  • Inspired by the social behavior of birds, which was studied by Craig Reynolds (a biologist) in late 80s and early 90s

  • Optimization problem representation is similar to the genes encoding methods used in GAs but for PSO the variables are called dimensions, that create a multi-dimensional hyperspace.

  • "Particles" fly in this hyperspace and try to find the global minima/maxima, their movement being governed by a simple mathematical equation.


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pi

xt

pg

xt+1

vt

PSO Basic Mathematical Equations

  • Basic mathematical equations in PSO:

particle’s personal best

particle’s neighbours best

where

particle’s itself


Repulsive pso 1 l.jpg

  • R1, R2, R3 : random numbers between 0 and 1ω : inertia weight between 0.01 and 0.7 : best position of a particleŷ : best position of a randomly chosen other particle from within the swarmz : a random velocity vectora, b, c : constants

Repulsive PSO (1)

  • RPSO is a global optimization algorithm, belongs to the class of stochastic evolutionary global optimizers, a variant of particle swarm optimization (PSO).


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Repulsive PSO (2)

  • The different realizations of PSO, where there is a repulsion between particles that can prevent the swarm being trapped in local minima (which would cause a premature convergence and would lead the optimization algorithm to fail to find the global optimum).

  • The main difference between PSO and RPSO is the propagation mechanism (vt+1) to determine new positions for a particle in the search space.

  • RPSO is capable of finding global optima in more complex search spaces. On the other hand, compared to PSO it may be slower on certain types of optimization problems.


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PSO Pseudocode

fori = 1 to number of particles n

forj =1 to number of dimensions m

C2 = uniform random number

C3 = uniform random number

V[ i ][ j ] = C1*V[ i ][ j ] + C2*(P[ i ][ j ]-X[ i ][ j ])

+ C3*(G[ i ][ j ]-X[ i ][ j ])

X[ i ][ j ] = X[ i ][ j ] + V[ i ][ j ]


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PSO Algorithms Common Parameter

  • c1/ω is an inertial constant. Good values are usually slightly less than 1.

  • c2 and c3 are two random vectors with each component generally a uniform random number between 0 and 1.

  • Very frequently the value of c1/ω is taken to decrease over time; e.g., one might have the PSO run for a certain number of iterations and decrease linearly from a starting value (0.9, say) to a final value (0.4, say) in order to facilitate exploitation over exploration in later states of the search.



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PSO to Continuous Problem

  • Continuous optimization problem as opposed to discrete optimization, the variables used in the objective function can assume real values, e.g., values fromintervals of the real line.

  • The particles "communicate" information they find about each other by updating their velocities in terms of local and global bests; when a new best is found, the particles will change their positions accordingly so that the new information is "broadcast" to the swarm.

  • The particles are always drawn back both to their own personal best positions and also to the best position of the entire swarm.

  • They also have stochastic exploration capability via the use of the random multipliers c2, and c3.

  • Typical convergence conditions include reaching a certain number of iterations, reaching a certain fitness value, and so on.


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PSO and Non-linear Maximization Problem

Non-linear Maximization problem:

f(x1,x2,x3) is maximum if

0 <= x1, x2, x3 <= 10

x1 = 10

x2 = 0

x3 = 10

f(x1,x2,x3) = 110


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Experiments and Results (1)

  • Both the PSO's and GA's approaches are implemented in Java v6.0 on Pentium4-1.80GHz CPU, 512M RAM, WinXP OS.

  • GA’s uses roulette wheel selection scheme, elitist model, one point crossover and uniform mutation.


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Experiments and Results (2)

GA’s Parameter

PSO’s Parameter


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Experiments and Results (3)

GA’s

Best max fitness value = 109.78

Best member:

x1 = 9.9931

x2 = 0.0075

x3 = 9.9949

Total time (ms) = 3469

PSO’s

Best max fitness value = 110.00

Best member:

x1 = 10.0

x2 = 0.0

x3 = 10.0

Total time (ms) = 344

Note:

Mean # of iteration = 72.540000

Mean fn val = 110.000000

Std. dev. fn val = 0.000000

Success rate = 100.00%


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Conclusion

  • PSO has proven both very effective and quick when applied to a diverse set of optimization problems.

  • GA’s results can be much better if uniform mutation, MU(x) := U([a,b]), is replaced by a Gaussian mutation, where x [a,b], m is mean, s is variance, andRi is sum of 12 random numbers from the range [0..1].

  • In future, it will be interesting to study and to compare the performance of PSO’s with GA’s and also ACO’s to solve discrete type of problem.


References l.jpg
References

  • Kennedy J, Eberhart R. C., and Shi Y. (2001). Swarm Intelligence. USA: Academic Press.

  • Michalewicz Z. (1996). Genetic Algorithms + Data Structures = Evolution Programs. 3rd, Revised and Extended Edition. USA: Springer.


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