psoptim {pso}

Particle Swarm Optimizer

Description

General implementation of particle swarm optimization usable as a direct replacement for optim.

Usage

psoptim(par, fn, gr = NULL, ..., lower = -1, upper = 1, control = list())

Arguments

par

Vector with length defining the dimensionality of the optimization problem. Providing actual values of par are not necessary (NA is just fine). Included primarily for compatibility with optim but if values are provided within the lower and upper bounds then the first particle will be initialized to the position provided by par.

fn

A function to be minimized (or maximized), with first argument the vector of parameters over which minimization is to take place. It should return a scalar result.

gr

A function to return the gradient if local search is BFGS. If it is NULL, a finite-difference approximation will be used.

...

Further arguments to be passed to fn and gr.

lower

Lower bounds on the variables.

upper

Upper bounds on the variables.

control

A list of control parameters. See “Details”.

Details

By default this function performs minimization using a particle swarm algorithm, but it will maximize if control$fnscale is negative.

The default control arguments implies that the algorithm follows the Standard PSO 2007 implementation by Maurice Clerc, but the code also provides support for PSO 2011, clamping the maximal velocity, restarting when all particles converge to a single area and using BFGS as the local search direction.

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

trace:

Non-negative integer. If positive, tracing information on the progress of the optimization is produced. Defaults to 0.

fnscale:

An overall scaling to be applied to the value of fn and gr (if used) during optimization. If negative, turns the problem into a maximization problem. Optimization is performed on fn(par)/fnscale. Defaults to 1.

maxit:

The maximum number of iterations. Defaults to 1000.

maxf:

The maximum number of function evaluations (not considering any performed during numerical gradient computation). Defaults to Inf.

abstol:

The absolute convergence tolerance. The method converges once the best fitness obtained is less than or equal to abstol. Defaults to -Inf.

reltol:

The tolerance for restarting. Once the maximal distance between the best particle and all other particles is less than reltol*d the algorithm restarts. Defaults to 0 which disables the check for restarting.

REPORT:

The frequency for reports if control$trace is positive. Defaults to 10.

trace.stats:

Logical; if TRUE statistics at every reporting step are collected and returned. Defaults to FALSE.

s:

The swarm size. Defaults to floor(10+2*sqrt(length(par))) unless type is “SPSO2011” in which case the default is 40.

k:

The exponent for calculating number of informants. Defaults to 3.

p:

The average percentage of informants for each particle. A value of 1 implies that all particles are fully informed. Defaults to 1-(1-1/s)^k.

w:

The exploitation constant. A vector of length 1 or 2. If the length is two, the actual constant used is gradially changed from w[1] to w[2] as the number of iterations or function evaluations approach the limit provided. Defaults to 1/(2*log(2)).

c.p:

The local exploration constant. Defaults to .5+log(2).

c.g:

The global exploration constant. Defaults to .5+log(2).

d:

The diameter of the search space. Defaults to the euclidean distance between upper and lower.

v.max:

The maximal (euclidean) length of the velocity vector. Defaults to NA which disables clamping of the velocity. However, if specified the actual clamping of the length is v.max*d.

rand.order:

Logical; if TRUE the particles are processed in random order. If vectorize is TRUE then the value of rand.order does not matter. Defaults to TRUE.

max.restart:

The maximum number of restarts. Defaults to Inf.

maxit.stagnate:

The maximum number of iterations without improvement. Defaults to Inf.

vectorize:

Logical; if TRUE the particles are processed in a vectorized manner. This reduces the overhead associated with iterating over each particle and may be more time efficient for cheap function evaluations. Defaults to FALSE.

hybrid:

If true, each normal PSO position update is followed by an L-BFGS-B search with the provided position as initial guess. This makes the implementation a hybrid approach. Defaults to FALSE which disables BFGS for the local search. Note that no attempt is done to control the maximal number of function evaluations within the local search step (this can be done separately through hybrid.control) but the number of function evaluations used by the local search method counts towards the limit provided by maxf AFTER the local search returns. To support a broader class of hybrid approaches a character vector can also be supplied with “off” being equivalent to false, “on” equivalent to true, and “improved” implying that the local search will only be performed when the swarm finds an improvement.

hybrid.control:

List with any additional control parameters to pass on to optim when using L-BFGS-B for the local search. Defaults to NULL.

type:

Character vector which describes which reference implementation of SPSO is followed. Can take the value of “SPSO2007” or “SPSO2011”. Defaults to “SPSO2007”.

Value

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

par

The best set of parameters found.

value

The value of fn corresponding to par.

counts

A three-element vector containing the number of function evaluations, the number of iterations, and the number of restarts.

convergence

An integer code. 0 indicates that the algorithm terminated by reaching the absolute tolerance; otherwise:

1:

Maximal number of function evaluations reached.

2:

Maximal number of iterations reached.

3:

Maximal number of restarts reached.

4:

Maximal number of iterations without improvement reached.

message

A descriptive message of the reason for termination.

If trace is positive and trace.stats is TRUE additionally the component:

stats

A list of statistics collected at every reporting step with the following components:

it

A vector with the iteration numbers

error

A vector with the corresponding best fitness values obtained

f

A list with the corresponding current swarm fitness values as a vector

x

A list with the corresponding current swarm positions as a matrix

References

Default parameters follow:

Clerc, M. (2011) https://hal.archives-ouvertes.fr/hal-00764996/document. Notice that the SPSO 2011 implementation does not include any of the bells and whistles from the implementation by M. Clerc et al. and effectively only differes from the SPSO 2007 implementation in the default swarm size, how velocities are initiated and the update of velocities/positions which in the SPSO 2011 implementation are invariant to rotation.

The gradual change of w and clamping the maximal velocity is described in:

Parsopoulos, K.E. and Vrahatis M.N. (2002) Recent approaches to global optimization problems through Particle Swarm Optimization. Natural Computing 1: 235-306.

The restart (provided through reltol) is similar to:

Evers G.I. and Ghalia M.B. Regrouping Particle Swarm Optimization: A New Global Optimization Algorithm with Improved Performance Consistency Across Benchmarks. https://bee22.com/resources/Evers%202009.pdf

The hybrid approach is similar to:

Qin J., Yin Y. and Ban X. (2010) A Hybrid of Particle Swarm Optimization and Local Search for Multimodal Functions. Lecture Notes in Computer Science, Volume 6145/2010, 589-596, DOI: 10.1007/978-3-642-13495-1_72

See Also

optim, test.problem.

Examples

set.seed(1)
## Rastrigin function
psoptim(rep(NA,2),function(x) 20+sum(x^2-10*cos(2*pi*x)),
        lower=-5,upper=5,control=list(abstol=1e-8))

set.seed(1)
## Rastrigin function - local refinement with L-BFGS-B on improvements
psoptim(rep(NA,2),function(x) 20+sum(x^2-10*cos(2*pi*x)),
        lower=-5,upper=5,control=list(abstol=1e-8,hybrid="improved"))

## Griewank function
psoptim(rep(NA,2),function(x) sum(x*x)/4000-prod(cos(x/sqrt(1:2)))+1,
        lower=-100,upper=100,control=list(abstol=1e-2))

set.seed(1)
## Rastrigin function with reporting
o <- psoptim(rep(NA,2),function(x) 20+sum(x^2-10*cos(2*pi*x)),
             lower=-5,upper=5,control=list(abstol=1e-8,trace=1,REPORT=1,
             trace.stats=TRUE))
## Not run: 
plot(o$stats$it,o$stats$error,log="y",xlab="It",ylab="Error")
points(o$stats$it,sapply(o$stats$f,min),col="blue",pch=2)

## End(Not run)