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Particle Swarm Optimization

This directory contains a simple implementation of particle swarm optimization (PSO.m), as well as scripts that use it to solve standard optimization test problems (TEST_PSO_*.m).

This implementation of PSO is designed for solving a bounded non-linear paramter optimization problem, with an initial guess. It is fully vectorized.

There are a variety of options that can be set by the user, but will be initialized to a default value if ommitted.

The output of the solver contains a full history of the optimization, which can be plotted using plotPsoHistory.m. Additionally, the user can define a plotting function to be called on each iteration.Both of these features are demonstrated in the TEST_PSO_*.m scripts.

The code supports both vectorized and non-vectorized objective function. If the objective function is vectorized, then the global best is updated synchronously, once per generation. If the objective function is not vectorized, then the optimization uses an asynchronous update, updating the global best after every particle update.

Test Functions:

  • TEST_PSO_1.m --> 2-D Sphere Function
  • TEST_PSO_2.m --> Himmelblau's function
  • TEST_PSO_3.m --> Goldstein-Price function
  • TEST_PSO_4.m --> 2-D Styblinski-Tang function
  • TEST_PSO_5.m --> N-D Styblinski-Tang function

Help file for PSO.m

[xBest, fBest, info, dataLog] = PSO(objFun, x0, xLow, xUpp, options)

Particle Swarm Optimization

This function minimizes OBJFUN using a variant of particle swarm
optimization. The optimization uses an initial guess X0, and searches
over a search space bounded by XLOW and XUPP.

INPUTS:
  objFun = objective function handle:
      f = objFun(x)
          x = [n, m] = search point in n-dimensional space (for m points)
          f = [1, m] = objective function value, for each of m points
  x0 = [n, 1] = initial search location
      --> Optional input. Set x0 = [] to ignore.
  xLow = [n, 1] = lower bounds on search space
  xUpp = [n, 1] = upper bounds on search space
  options = option struct. All fields are optional, with defaults:
      .alpha = 0.6 = search weight on current search direction
      .beta = 0.9 = search weight on global best
      .gamma = 0.9 = search weight on local best
      .nPopulation = m = 3*n = population count
      .maxIter = 100 = maximum number of generations
      .tolFun = 1e-6 = exit when variance in objective is < tolFun
      .tolX = 1e-10 = exit when norm of variance in state < tolX
      .flagVectorize = false = is the objective function vectorized?
      .flagMinimize = true = minimize objective
          --> Set to false to maximize objective
      .flagWarmStart = false = directly use initial guess?
          --> true:  first particle starts at x0
          --> false: all particles are randomly selected
      .guessWeight = 0.2;  trade-off for initialization; range [0, 0.9)
          --> 0.0  ignore x0; use random initialization [xLow, xUpp]
          --> 0.9  heavy weight on initial guess (x0)
      .plotFun = function handle for plotting progress
          plotFun( dataLog(iter), iter )
          --> See OUTPUTS for details about dataLog
          --> Leave empty to omit plotting (faster)
      .display = 'iter';
          --> 'iter' = print out info for each iteration
          --> 'final' = print out some info on exit
          --> 'off' = disable printing
      .printMod = 1   (only used if display == 'iter')

OUTPUTS:
  xBest = [n, 1] = best point ever found
  fBest = [1, 1] = value of best point found

  info = output struct with solver info
      .input = copy of solver inputs:
          .objFun
          .x0
          .xLow
          .xUpp
          .options
      .exitFlag = how did optimization finish
          0 = objective variance < tolFun
          1 = reached max iteration count
          2 = norm of state variance < tolX
      .fEvalCount = how many calls to the objective function?
      .X_Global = [n,iter] = best point in each generation
      .F_Global = [1,iter] = value of the best point ever
      .I_Global = [1,iter] = index of the best point ever
      .X_Best_Var = [n,iter] = variance in best point along each dim
      .X_Var = [n,iter] = variance in current search along each dim
      .X_Best_Mean = [n,iter] = mean in best point along each dim
      .X_Mean = [n,iter] = mean in current search along each dim
      .F_Best_Var = [1,iter] = variance in the best val at each gen
      .F_Var = [1,iter] = variance in the current val at each gen
      .F_Best_Mean = [1,iter] = mean of the population best value
      .F_Mean = [1,iter] = mean of the current population value

  dataLog(iter) = struct array with data from each iteration
      .X = [n,m] = current position of each particle
      .V = [n,m] = current "velocity" of each particle
      .F = [1,m] = value of each particle
      .X_Best = [n,m] = best point for each particle
      .F_Best = [1,m] = value of the best point for each particle
      .X_Global = [n,1] = best point ever (over all particles)
      .F_Global = [1,1] = value of the best point ever
      .I_Global = [1,1] = index of the best point ever

NOTES:
  This function uses a slightly different algorithm based on whether or
  not the objective function is vectorized. If the objective is
  vectorized, then the new global best point is only computed once per
  iteration (generation). If the objective is not vectorized, then the
  global best is updated after each particle is updated.


DEPENDENCIES
  --> mergeOptions()
  --> makeStruct()


REFERENCES:

  http://www.scholarpedia.org/article/Particle_swarm_optimization

  Clerc and Kennedy (2002)