MOPSO : A Proposal for Multiple Objective Particle Swarm Optimization

1
MOPSO A Proposal for Multiple Objective
Particle Swarm Optimization

• Computational Intelligence 2002
• Carlos A. Coello Coello
• Maximino Salazar Lechuga
• Pruet _at_ DSSG

2
Outline

• Introduction
• Multi-Objective Particle Swarm Optimization
  (MOPSO)
• Evaluation

Bird Flocking, an example of natural swarming
Figure from http//www.aip.org/png/html/bird.htm
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Introduction

• To increase the efficiency of multi-objective
  optimization algorithm
• diversity of the result has to be maintained
• use of small population sizes.
• Particle swarm optimization (PSO) is a heuristic
  inspired by the choreography of a bird flock.
• Most of the works in PSO focus on
  single-objective optimization.
• The authors propose an algorithm called multi
  objective particle swarm optimization (MOPSO)
• MOPSO allows the PSO algorithm to be able to deal
  with multiobjective optimization problems.

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Introduction

• Particle Swarm Optimization (PSO)
• The PSO can bee seen as a distributed behavioral
  algorithm performs muldimentional search.
• The behavior of each individual is affected by
  either the local best or the global best.
• The approach uses the concept of population and a
  measure of performance similar to the fitness
  function used in EA.
• Moreover, the adjustments of individuals are
  analogous to the use of crossover operators.
• However, PSO introduces the use of flying
  potential solution through hyperspace (used to
  accelerate convergence).
• Also, PSO allows individual to benefit from their
  past experiences, in contrast with EA, which
  focus on present experience, i.e. recent
  population pool.
• Elitism or Archive can be seen as approaches to
  preserve the past best, but they are pool-wise,
  not individual-wise.

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MOPSO

• In this paper, the concept of Pareto ranking
  scheme is used for measuring the efficiency of
  multiobjective optimization algorithm.
• The authors suggest that the use of Pareto in
  multi-objective PSO makes sense because of the
  similarity between PSO and EA.
• Multiple Objective Particle Swarm Optimization
  (MOPSO)
• The historical record of best solution found by a
  particle could be used to store nondominated
  solutions generated in the past.
• Therefore, the author propose the idea of having
  a global repository in which every particle will
  deposit it flight experiences after each flight
  cycle.
• Additionally, the updates to the repository are
  performed considering a geographically-based
  system defined in terms of the objective function
  values of each individual.
• To maintain diversity
• This technique is inspired from PAES (Pareto
  Archive Evolution Strategy)
• The repository is used by the particles to
  identify a leader that will guide the search.
• random search should be added to avoid local
  optimum.

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MOPSO Algorithm

• Initialize the population POP
• FOR i 0 to MAX / MAX population size /
• initialize POPi / POPi is the position
  of particle i in search space /
• Initialize the speed of each particle
• FOR i 0 to MAX
• VELi 0
• Evaluate each of the particles in POP
• Store the positions of the particles that is
  nondominated in the repository REP.
• Generate hypercube of the search space explored
  so far, and locate the particles using these
  hypercube as a coordinate system (each axis
  correspond to an objective function)
• Initialize the memory of each particle (this
  memory serves as a guide to travel through the
  search space.)
• FOR i 0 to MAX
• PBESTi PIPI

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MOPSO Algorithm

• WHILE maximum number of cycles has not been reach
  DO
• Compute the speed of each particle
• W is inertia weight (less than 1)
• R1 and R2 are random numbers in the range of
  0..1
• PBESTi is the best position so far of particle
  i
• REPh is a value from repository, h is selected
  by
• those hypercube containing more than one
  particle are assigned a fitness equal to the
  result of diving any number x gt 1 by the number
  of particles that they contain. Then, the result
  is used in roulette wheel selection to select a
  hypercube. A particle in the selected hypercube
  is selected randomly as h.
• POPi is the current value of the particle i
• Compute the new position of each particle
• Maintain (remove ?) particles which go outside
  search space

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MOPSO Algorithm

• Evaluate each of the particles in POP
• Update the contents of REP togeher with the
  content of hypercube.
• Insert nondominated particles into the
  repository.
• Remove all dominated particles in the repository.
• If the repository is full, the particles in
  crowed hypercube are removed.
• When the current position of the particle is
  better than the the position in its PBEST, the
  particle updates PBESTi POPi
• better in term of dominance.

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Evaluation

• Two matrices
• Average running time of algorithm.
• Average distance to Pareto optimal set
• Y,Y ? Y are the set of objective vectors that
  correspond to a set of pairwise nondominating
  decision vectors X,X ? X. X corresonds to the
  decision variable of the problem.
• Compared with
• NSGA II
• Population size 200, cross over rate 0.8, use
  tournament selection, and mutation rate
  1/(number of decision variables of the problem)
• PAES
• Archive size 200, mutation rate 1/L L
  length of the chromosome
• MOPSO
• Population size 40, repository size 200, 30
  division per grid.

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Test Function 1

• Average running time (4000 cycles)
• NSGA II 2.402
• PAES 2.08
• MOPSO 0.076
• M1 avr/sd
• NSGA 0.002536/0.000138
• PAES 0.002881/0.00213
• MOPSO 0.002057/0.000286

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Test Function 2

• Average running time (1200 cycles)
• NSGA II 0.812
• PAES 0.339
• MOPSO 0.046
• M1 avr/sd
• NSGA 0.001594/0.000122
• PAES 0.070003/0.158081
• MOPSO 0.00147396/0.00020178

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Test Function 3

• Average running time (3200 cycles)
• NSGA II 2.1165
• PAES 1.641
• MOPSO 0.098
• M1 avr/sd
• NSGA 0.094644/0.117608
• PAES 0.259664/0.57386
• MOPSO 0.0011611/0.0007205

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Conclusion

• MOPSO performed reasonably well in terms of the
  average distance to Pareto front, with lower
  computational times.
• PSO is an unconstrained search technique, it is
  necessary to develop an additional mechanism to
  deal with constrained multiobjective optimization
  problems.