Particle Swarm Optimization Algorithms to Continuous Problem

1
Particle Swarm Optimization Algorithms to
Continuous Problem
Sunday, November 15, 2009 by

• Yoon-Teck Bau, Hong-Tat Ewe, Chin-Kuan Ho
• Faculty of Information Technology
• Multimedia University, Malaysia
• ytbau, htewe, ckho_at_mmu.edu.my
• http//pesona.mmu.edu.my/ytbau/

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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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PSO Basic Mathematical Equations

• Basic mathematical equations in PSO

particles personal best
particles neighbours best
where
particles itself
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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 (vt1) 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

• for i 1 to number of particles n
• for j 1 to number of dimensions m
• C2 uniform random number
• C3 uniform random number
• V i j C1V 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/? 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/? 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
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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 from intervals 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 lt x1, x2, x3 lt 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.
• GAs uses roulette wheel selection scheme,
  elitist model, one point crossover and uniform
  mutation.

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GAs Parameter
Experiments and Results (2)
PSOs Parameter
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Experiments and Results (3)
GAs Best max fitness value 109.78 Best
member x1 9.9931 x2 0.0075 x3
9.9949 Total time (ms) 3469

• PSOs
• 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.
• GAs 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, and
  Ri 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 PSOs with GAs and
  also ACOs to solve discrete type of problem.

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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.