hydroPSO (version 0.5-1)

hydroPSO: Enhanced Particle Swarm Optimisation algorithm

Description

State-of-the-art version of the Particle Swarm Optimisation (PSO) algorithm (SPSO-2011 and SPSO-2007 capable). hydroPSO can be used as a replacement for optim, but its main focus is the calibration of environmental and other real-world model codes. Several fine-tuning options and PSO variants are available to customise the PSO engine to different calibration problems.

Usage

hydroPSO(par, fn= "hydromod", ..., 
         method=c("spso2011", "spso2007", "ipso", "fips", "wfips", "canonical"),
         lower=-Inf, upper=Inf, control=list(), 
         model.FUN=NULL, model.FUN.args=list() )

Arguments

par

OPTIONAL. numeric with a first guess for the parameters to be optimised, with length equal to the dimension of the solution space All the particles are randomly initialised according to the value of Xini.type. If the user provides m parameter sets for par, they are used to overwrite the first m parameter sets randomly defined according to the value of Xini.type. If some elements in par are non finite (lower than lower or larger than upper) they are ignored

fn

function or character with the name of a valid R function to be optimised (minimized or maximized). The character value ‘hydromod’ is used to specify that an R-external model code (i.e., an executable file that needs to be run from the system console) will be analised instead of an R function -) When fn!='hydromod', the first argument of fn has to be a vector of parameters over which optimisation is going to take place. It should return a scalar result. When fn!='hydromod' the algorithm uses the value(s) returned by fn as both model output and its corresponding goodness-of-fit measure -) When fn=='hydromod' the algorithm will optimise the model defined by model.FUN and model.args, which are used to extract the values simulated by the model and to compute its corresponding goodness-of-fit measure

OPTIONAL. Only used when fn!='hydromod' & fn!='hydromodInR'. further arguments to be passed to fn.

method

character, variant of the PSO algorithm to be used. By default method='spso2011', while valid values are ‘spso2011’, ‘spso2007’, ‘ipso’, ‘fips’, ‘wfips’, ‘canonical’:

spso2011: At each iteration particles are attracted to its own best-known ‘personal’ and to the best-known position in its ‘local’ neighbourhood, which depens on the value of topology. In addition, values of the PSO engine are set to the values defined in the Standard PSO 2011 (SPSO 2011, see Clerc 2012)

spso2007: As in method='spso2011', but with values of the PSO engine set to the values defined in the Standard PSO 2007 (SPSO 2007, see Clerc 2012)

ipso: at each iteration particles in the swarm are rearranged in descending order according to their goodness-of-fit and the best ngbest particles are used to modify particles' position and velocity (see Zhao, 2006). Each particle is connected to a neighbourhood of particles depending on the topology value

fips: at each iteration ALL particles contribute to modify the particles' position and velocity (see Mendes et al., 2004). Each particle is connected to a neighbourhood of particles depending on the topology value

wfips: same implementation as fips method, but the contribution of each particle is weighted according to their goodness-of-fit value (see Mendes et al., 2004)

canonical: It corresponds to the first formulation of the PSO algorithm, and it is included here for educational and comparative purposes only, due to several limitations described in literature (see Kennedy 2006). At each iteration, particles are attracted to its own best-known ‘personal’ and to the best-known position in all the swarm (‘global’). The following control arguments are set when this method is selected: (i) npart=40, (ii) topology='gbest', (iii) Xini.type='random', (iv) Vini.type='random2007', (v) use.CF=TRUE, (vi) c1=2.05, (vii) c2=2.05, (viii) boundary.wall='absorbing2007', (ix) lambda=1.0

lower

numeric, lower boundary for each parameter Note for optim users: in hydroPSO the length of lower and upper are used to defined the dimension of the solution space

upper

numeric, upper boundary for each parameter Note for optim users: in hydroPSO the length of lower and upper are used to defined the dimension of the solution space

control

a list of control parameters. See ‘Details’

model.FUN

OPTIONAL. Used only when fn='hydromod' character, valid R function representing the model code to be calibrated/optimised

model.FUN.args

OPTIONAL. Used only when fn='hydromod' list with the arguments to be passed to model.FUN

Value

A list, compatible with the output from optim, with components:

par

optimum parameter set found

value

value of fn corresponding to par

counts

three-element vector containing the total number of function calls, number of iterations, and number of regroupings

convergence

integer code where 0 indicates that the algorithm terminated by reaching the absolute tolerance, otherwise:

1

relative tolerance reached

2

maximum number of (effective) function evaluations reached

3

maximum number of iterations reached

message

character string giving human-friendly information about convergence

Details

By default the hydroPSO function performs minimization of fn, but it will maximize fn if MinMax='max'

The default control arguments in hydroPSO implements the Standard PSO 2011 - SPSO2011 (see Clerc 2012; Clerc et al., 2010). At the same time, hydroPSO function provides options for clamping the maximal velocity, regrouping strategy when premature convergence is detected, time-variant acceleration coefficients, time-varying maximum velocity, (non-)linear / random / adaptive / best-ratio inertia weight definitions, random or LHS initialization of positions and velocities, synchronous or asynchronous update, 4 alternative neighbourhood topologies among others

The control argument is a list that can supply any of the following components:

drty.in

OPTIONAL. Used only when fn='hydromod' character, name of the directory storing the input files required for PSO, i.e. ‘ParamRanges.txt’ and ‘ParamFiles.txt’

drty.out

character, path to the directory storing the output files generated by hydroPSO

param.ranges

OPTIONAL. Used only when fn='hydromod' character, name of the file defining the minimum and maximum boundary values for each one of the parameters to be calibrated

digits

OPTIONAL. Used only when write2disk=TRUE numeric, number of significant digits used for writing the output files with scientific notation

MinMax

character, indicates whether a maximization or minimization problem needs to be solved. Valid values are in: c('min', 'max'). Default value is min

npart

numeric, number of particles in the swarm. By default npart=NA, which means that the swarm size depends on the value of method: when method='spso2007' npart=ceiling(10+2*sqrt(n)), or npart=40 otherwise

maxit

numeric, maximum number of iterations. By default maxit=1000

maxfn

numeric, maximum number of function evaluations. Default value is +Inf When fn=='hydromod', this stopping criterion uses the number of effective function calls, i.e. those function calls with a finite output value

c1

numeric, cognitive acceleration coefficient. Encourages the exploitation of the solution space and reflects how much the particle is influenced by its own best-known position By default c1= 0.5 + log(2)

c2

numeric, social acceleration coefficient. Encourages the exploration of the current global best and reflects how much the particle is influenced by the best-known optimum of the swarm By default c2= 0.5 + log(2)

use.IW

logical, indicates if an inertia weight (w) will be used to avoid swarm explosion, i.e. particles flying around their best position without converging into it (see Shi and Eberhart, 1998) By default use.IW=TRUE

IW.w

OPTIONAL. Used only when use.IW= TRUE \& IW.type!='GLratio' numeric, value of the inertia weight(s) (w or [w.ini, w.fin]). It can be a single number which is used for all iterations, or it can be a vector of length 2 with the initial and final values (in that order) that w will take along the iterations By default IW.w=1/(2*log(2))

use.CF

logical, indicates if the Clerc's Constriction Factor (see Clerc, 1999; Eberhart and Shi, 2000; Clerc and Kennedy, 2002) is used to avoid swarm explosion By default use.CF=FALSE

lambda

numeric in [0,1], represents a percentage to limit the maximum velocity (Vmax) for each dimension, which is computed as vmax = lambda*(Xmax-Xmin) By default lambda=1

abstol

numeric, absolute convergence tolerance. The algorithm stops if gbest <= abstol (minimisation problems) OR when gbest >= abstol (maximisation problems) By default it is set to -Inf or +Inf for minimisation or maximisation problems, respectively

reltol

numeric, relative convergence tolerance. The algorithm stops if the absolute difference between the best ‘personal best’ in the current iteration and the best ‘personal best’ in the previous iteration is less or equal to reltol. Defaults to sqrt(.Machine$double.eps), typically, about 1e-8 If reltol is set to 0, this stopping criterion is not used

Xini.type

character, indicates how to initialise the particles' positions in the swarm within the ranges defined by lower and upper. Valid values are: -) lhs: Latin Hypercube initialisation of positions, using npart number of strata to divide each parameter range. It requires the lhs package -) random: random initialisation of positions within lower and upper By default Xini.type='random'

Vini.type

character, indicates how to initialise the particles' velocities in the swarm. Valid values are: -) random2011: random initialisation of velocities within lower-Xini and upper-Xini, as defined in SPSO 2011 (Vini=U(lower-Xini, upper-Xini)) (see Clerc, 2012, 2010) -) lhs2011: same as in random2011, but using a Latin Hypercube initialisation with npart number of strata instead of a random uniform distribution for each parameter. It requires the lhs package -) random2007: random initialisation of velocities within lower and upper using the ‘half-diff’ method defined in SPSO 2007 (Vini=[U(lower, upper)-Xini]/2) (see Clerc, 2012, 2010) -) lhs2007: same as in random2007, but using a Latin Hypercube initialisation with npart number of strata instead of a random uniform distribution for each parameter. It requires the lhs package -) zero: all the particles are initialised with zero velocity By default Vini.type=NA, which means that Vini.type depends on the value of method: when method='spso2007' Vini.type='random2007', or Vini.type='random2011' otherwise

best.update

character, indicates how (when) to update the global/neighbourhood and personal best. Valid values are: -)sync: the update is made synchronously, i.e. after computing the position and goodness-of-fit for ALL the particles in the swarm. This is the DEFAULT option -)async: the update is made asynchronously, i.e. after computing the position and goodness-of-fit for EACH individual particle in the swarm

random.update

OPTIONAL. Only used when best.update='async' logical, if TRUE the particles are processed in random order to update their personal best and the global/neighbourhood best By default random.update=TRUE

boundary.wall

character, indicates the type of boundary condition to be applied during optimisation. Valid values are: NA, ‘absorbing2011’, ‘absorbing2007’, ‘reflecting’, ‘damping’, ‘invisible’ By default boundary.wall=NA, which means that boundary.wall depends on the value of method: when method='spso2007' boundary.wall='absorbing2007', or boundary.wall='absorbing2011' otherwise

Experience has shown that Clerc's constriction factor and the inertia weights do not always confine the particles within the solution space. To address this problem, Robinson and Rahmat-Samii (2004) and Huang and Mohan (2005) propose different boundary conditions, namely, reflecting, damping, absorbing and invisible to define how particles are treated when reaching the boundary of the searching space (see Robinson and Rahmat-Samii (2004) and Huang and Mohan (2005) for further details)

topology

character, indicates the neighbourhood topology used in hydroPSO. Valid values are in c('random', 'gbest', 'lbest', 'vonNeumann'): -) gbest: every particle is connected to each other and, hence the global best influences all particles in the swarm. This is also termed star topology, and it is generally assumed to have a fast convergence but is more vulnerable to the attraction to sub-optimal solutions (see Kennedy, 1999; Kennedy and Mendes, 2002, Schor et al., 2010) -) lbest: each particle is connected to its K immediate neighbours only. This is also termed circles or ring topology, and generally the swarm will converge slower than the gbest topology but it is less vulnerable to sub-optimal solutions (see Kennedy, 1999; Kennedy and Mendes, 2002) -) vonNeumann: each particle is connected to its K=4 immediate neighbours only. This topology is more densely connected than lbest but less densely than gbest, thus, showing some parallelism with lbest but benefiting from a bigger neighbourhood (see Kennedy and Mendes, 2003) -) random: the random topology is a special case of lbest where connections among particles are randomly modified after an iteration showing no improvement in the global best (see Clerc, 2005; Clerc, 2010) By default topology='random'

K

OPTIONAL. Only used when topology is in c(random, lbest, vonNeumann) numeric, neighbourhood size, i.e. the number of informants for each particle (including the particle itself) to be considered in the computation of their personal best When topology=lbest K MUST BE an even number in order to consider the same amount of neighbours to the left and the right of each particle As special case, K could be equal to npart. By default K=3

iter.ini

OPTIONAL. Only used when topology=='lbest' numeric, number of iterations for which the gbest topology will be used before using the lbest topology for the computation of the personal best of each particle This option aims at making faster the identification of the global zone of attraction By default iter.ini=0

ngbest

OPTIONAL. Only used when method=='ipso' numeric, number of particles considered in the computation of the global best By default ngbest=4 (see Zhao, 2006)

normalise

logical, indicates whether the parameter values have to be normalised to the [0,1] interval during the optimisation or not This option appears in the C and Matlab version of SPSO-2011 (See http://www.particleswarm.info/standard_pso_2011_c.zip) and there it is recommended to use this option when the search space is not an hypercube. If the search space is an hypercube, it is better not normalise (there is a small difference between the position without any normalisation and the de-normalised one). By default normalise=FALSE

IW.type

OPTIONAL. Used only when use.IW= TRUE AND length(IW.w)>1 character, defines how the inertia weight w will vary along iterations. Valid values are: -)linear: w varies linearly between the initial and final values specified in IW.w (see Shi and Eberhart, 1998; Zheng et al., 2003). This is the DEFAULT option -)non-linear: w varies non-linearly between the initial and final values specified in IW.w with exponential factor IW.exp (see Chatterjee and Siarry, 2006) -)runif: w is a uniform random variable in the range [w.min, w.max] specified in IW.w. It is a generalisation of the weight proposed in Eberhart and Shi (2001b) -)aiwf: adaptive inertia weight factor, where the inertia weight is varied adaptively depending on the goodness-of-fit values of the particles (see Liu et al., 2005) -)GLratio: w varies according to the ratio between the global best and the average of the particle's local best (see Arumugam and Rao, 2008) By default IW.type='linear'

IW.exp

OPTIONAL. Used only when use.IW=TRUE AND IW.type='non-linear' numeric, non-linear modulation index (see Chatterjee and Siarry, 2006) When IW.type='linear', IW.exp is set to 1. By default IW.exp=1

use.TVc1

logical, indicates if the cognitive acceleration coefficient c1 will have a time-varying value instead of a constant one provided by the user (see Ratnaweera et al. 2004). By default use.TVc1=FALSE

TVc1.type

character, required only when use.TVc1 = TRUE. Valid values are: -)linear: c1 varies linearly between the initial and final values specified in TVc1.rng (see Ratnaweera et al., 2004) -)non-linear: c1 varies non-linearly between the initial and final values specified in TVc1.rng. Proposed by the authors of hydroPSO taking into account the work of Chatterjee and Siarry (2006) for the inertia weight -)GLratio: c1 varies according to the ratio between the global best and the average of the particle's local best (see Arumugam and Rao, 2008) By default TVc1.type='linear'

TVc1.rng

OPTIONAL. Used only when use.TVc1=TRUE AND TVc1.type!='GLratio' numeric, initial and final values for the cognitive acceleration coefficient [c1.ini, c1.fin] (in that order) along the iterations By default TVc1.rng=c(1.28, 1.05)

TVc1.exp

OPTIONAL. Used only when use.TVc1= TRUE AND TVc1.type= 'non-linear' numeric, non-linear modulation index When TVc1.exp is equal to 1, TVc1 corresponds to the improvement proposed by Ratnaweera et al., (2004), whereas when TVc1.exp is different from one, no reference has been found in literature by the authors, but it was included as an option based on the work of Chatterjee and Siarry (2006) for the inertia weight When TVc1.type='linear', TVc1.exp is automatically set to 1. By default TVc1.exp=1

use.TVc2

logical, indicates whether the social acceleration coefficient c2 will have a time-varying value or a constant one provided by the user (see Ratnaweera et al. 2004). By default use.TVc2=FALSE

TVc2.type

character, required only when use.TVc2=TRUE. Valid values are: -)linear: c2 varies linearly between the initial and final values specified in TVc2.rng (see Ratnaweera et al. 2004) -)non-linear: c2 varies non-linearly between the initial and final values specified in TVc2.rng. Proposed by the authors of hydroPSO taking into account the work of Chatterjee and Siarry (2006) for the inertia weight By default TVc2.type='linear'

TVc2.rng

OPTIONAL. Used only when use.TVc2=TRUE numeric, initial and final values for the social acceleration coefficient [c2.ini, c2.fin] (in that order) along the iterations By default TVc2.rng=c(1.05, 1.28)

TVc2.exp

OPTIONAL. Used only when use.TVc2= TRUE AND TVc2.type='non-linear' numeric, non-linear modulation index When TVc2.exp is equal to 1, TVc2 corresponds to the improvement proposed by Ratnaweera et al., 2004, whereas when TVc2.exp is different from one, no reference has been found in literature by the authors, but it was included as an option based on the work of Chatterjee and Siarry (2006) for the inertia weight When TVc2.type= linear, TVc2.exp is automatically set to 1. By default TVc2.exp=1

use.TVlambda

logical, indicates whether the percentage to limit the maximum velocity lambda will have a time-varying value or a constant value provided by the user. Proposed by the authors of hydroPSO based on the work of Chatterjee and Siarry (2006) for the inertia weight By default use.TVlambda=FALSE

TVlambda.type

character, required only when use.TVlambda=TRUE. Valid values are: -)linear: TVvmax varies linearly between the initial and final values specified in TVlambda.rng -)non-linear: TVvmax varies non-linearly between the initial and final values specified in TVlambda.rng By default TVlambda.type='linear'

TVlambda.rng

OPTIONAL. Used only when use.TVlambda=TRUE numeric, initial and final values for the percentage to limit the maximum velocity [TVlambda.ini, TVlambda.fin] (in that order) along the iterations By default TVlambda.rng=c(1, 0.25)

TVlambda.exp

OPTIONAL. only required when use.TVlambda= TRUE AND TVlambda.type='non-linear' numeric, non-linear modulation index When TVlambda.type='linear', TVlambda.exp is automatically set to 1. By default TVlambda.exp=1

use.RG

logical, indicates if the swarm should be regrouped when premature convergence is detected. By default use.RG=FALSE When use.RG=TRUE the swarm is regrouped in a search space centred around the current global best. This updated search space is hoped to be both small enough for efficient search and large enough to allow the swarm to escape from stagnation (see Evers and Ghalia, 2009) There are 4 differences wrt Evers and Ghalia 2009: -) swarm radius: median is used instead of max -) computation of the new range of parameter space, which corresponds to the boundaries of the whole swarm at a given iteration, instead of the maximum values of ‘abs(x-Gbest)’ -) regrouping factor: RG.r instead of ‘6/(5*ro)’ -) velocity is re-initialized using Vini.type instead of using the formula proposed by Evers and Ghalia 2009

RG.thr

ONLY required when use.RG=TRUE numeric, positive number representing the stagnation threshold used to decide whether the swarm has to be regrouped or not. See Evers and Galia (2009) for further details Regrouping occurs when the normalised swarm radius is less than RG.thr. By default RG.thr=1E-5

RG.r

ONLY required when use.RG=TRUE. numeric, positive number representing the regrouping factor, which is used to regroup the swarm in a search space centred around the current global best (see Evers and Galia, 2009 for further details). By default RG.thr=2

RG.miniter

ONLY required when use.RG=TRUE numeric, minimum number of iterations needed before each new regrouping. By default RG.miniter=100

%% \item{use.DS}{ %%CPSO %%} %% \item{DS.r}{ %% ~~Describe \code{DS.r} here~~ %%} %% \item{DS.tol}{ %% ~~Describe \code{DS.tol} here~~ %%} %% \item{DS.dmin}{ %% ~~Describe \code{DS.dmin} here~~ %%}
plot

logical, indicates if a two-dimensional plot with the particles' position will be drawn after each iteration. For high dimensional functions, only the first two dimensions of all the particles are plotted By default plot=FALSE

out.with.pbest

logical, indicates if the best parameter values for each particle and their goodness-of-fit will be included in the output of the algorithm By default out.with.pbest=FALSE

out.with.fit.iter

logical, indicates if the goodness-of-fit of each particle for each iteration will be included in the output of the algorithm By default out.with.fit.iter=FALSE

write2disk

logical, indicates if the output files will be written to the disk. By default write2disk=TRUE

verbose

logical, indicates if progress messages are to be printed. By default verbose=TRUE

REPORT

OPTIONAL. Used only when verbose=TRUE The frequency of report messages printed to the screen. Default to every 100 iterations

parallel

character, indicates how to parallelise ‘hydroPSO’ (to be precise, only the evaluation of the objective function fn is parallelised). Valid values are: -)none: no parallelisation is made (this is the default value) -)multicore: DEPRECATED!, since multicore package is not in CRAN anymore. Originally it was thought to carry out parallel computations for machines with multiple cores or CPUs. The evaluation of the objective function fn is done with the mclapply function of the parallel package. It requires POSIX-compliant OS (essentially anything but Windows) -)parallel: parallel computations for network clusters or machines with multiple cores or CPUs. A ‘FORK’ cluster is created with the makeForkCluster function. When fn.name="hydromod" the evaluation of the objective function fn is done with the clusterApply function of the parallel package. When fn.name!="hydromod" the evaluation of the objective function fn is done with the parRapply function of the parallel package. -)parallelWin: parallel computations for network clusters or machines with multiple cores or CPUs (this is the only parallel implementation that works on Windows machines). A ‘PSOCK’ cluster is created with the makeCluster function. When fn.name="hydromod" the evaluation of the objective function fn is done with the clusterApply function of the parallel package. When fn.name!="hydromod" the evaluation of the objective function fn is done with the parRapply function of the parallel package.

par.nnodes

OPTIONAL. Used only when parallel!='none' numeric, indicates the number of cores/CPUs to be used in the local multi-core machine, or the number of nodes to be used in the network cluster. By default par.nnodes is set to the amount of cores detected by the function detectCores() (multicore or parallel package)

par.pkgs

OPTIONAL. Used only when parallel='parallelWin' list of package names (as characters) that need to be loaded on each node for allowing the objective function fn to be evaluated

References

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See Also

optim

Examples

Run this code
# NOT RUN {
# Number of dimensions of the optimisation problem (for all the examples)
D <- 5

# Boundaries of the search space (Rastrigin function)
lower <- rep(-5.12, D)
upper <- rep(5.12, D)

# }
# NOT RUN {
################################ 
# Example 1. Basic use         #
################################ 

# Setting the seed (for reproducible results)         
set.seed(100)

# Basic use 1. Rastrigin function (non-linear and multi-modal with many local minima)
# Results are not saved to the hard disk, for faster execution ('write2disk=FALSE')
hydroPSO(fn=rastrigin, lower=lower, upper=upper, control=list(write2disk=FALSE) )
# }
# NOT RUN {
  # donttest END

# }
# NOT RUN {
setwd("~")
# Basic use 2. Rastrigin function (non-linear and multimodal with many local minima)
# Results are saved to the hard disk. Slower than before but results are kept for
# future inspection
hydroPSO(fn=rastrigin, lower=lower, upper=upper )

# Plotting the results, by default into the active graphic device
# 'MinMax="min"' indicates a minimisation problem
plot_results(MinMax="min") 

# Plotting the results into PNG files. 
plot_results(MinMax="min", do.png=TRUE)         
# }
# NOT RUN {
 # dontrun END


# }
# NOT RUN {
################################ 
# Example 2. More advanced use #
################################ 

# Defining the relative tolerance ('reltol'), the frequency of report messages 
# printed to the screen ('REPORT'), and no output files ('write2disk')
set.seed(100)
hydroPSO( fn=rastrigin, lower=lower, upper=upper,        
          control=list(reltol=1e-20, REPORT=10, write2disk=FALSE) )
        
        
################################### 
# Example 3. von Neumman Topology #
###################################

# Same as Example 2, but using a von Neumann topology ('topology="vonNeumann"')
set.seed(100)
hydroPSO(fn=rastrigin,lower=lower,upper=upper,
         control=list(topology="vonNeumann", reltol=1E-20, 
                      REPORT=50, write2disk=FALSE) ) 



################################ 
# Example 4. Regrouping        #
################################ 

# Same as Example 3 ('topology="vonNeumann"') but using regrouping ('use.RG')
set.seed(100)
hydroPSO(fn=rastrigin,lower=lower,upper=upper,
         control=list(topology="vonNeumann", reltol=1E-20, 
                      REPORT=50, write2disk=FALSE,
                      use.RG=TRUE,RG.thr=7e-2,RG.r=3,RG.miniter=50) )


################################ 
# Example 5. FIPS              #
################################ 

# Same as Example 3 ('topology="vonNeumann"') but using a fully informed 
# particle swarm (FIPS) variant ('method') with global best topology
set.seed(100)
hydroPSO(fn=rastrigin,lower=lower,upper=upper, method="fips",
         control=list(topology="gbest",reltol=1E-9,write2disk=FALSE) )


################################ 
# Example 6. normalisation     #
################################ 

# Same as Example 3 but parameter values are normalised to the [0,1] interval 
# during the optimisation. This option is recommended when the search space is 
# not an hypercube (not useful is this particular example)
set.seed(100)
hydroPSO(fn=rastrigin,lower=lower,upper=upper,
         control=list(topology="vonNeumann", reltol=1E-20, normalise=TRUE,
                      REPORT=50, write2disk=FALSE) ) 


################################ 
# Example 7. Asynchronus update#
################################ 

# Same as Example 3, but using asynchronus update of previus and local best 
# ('best.update'). Same global optimum but much slower....
set.seed(100)
hydroPSO(fn=rastrigin,lower=lower,upper=upper,
         control=list(topology="vonNeumann", reltol=1E-20, 
                      REPORT=50, write2disk=FALSE, best.update="async") ) 

# }
# NOT RUN {
 # donttest END

# }

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