NMOF (version 0.20-0)

PSopt: Particle Swarm Optimisation

Description

The function implements Particle Swarm Optimisation.

Usage

PSopt(OF, algo = list(), ...)

Arguments

OF

the objective function to be minimised. See Details.

algo

a list with the settings for algorithm. See Details and Examples.

...

pieces of data required to evaluate the objective function. See Details.

Value

• Returns a list:
• xbest
  the solution
• OFvalue
  objective function value of best solution
• popF
  a vector: the objective function values in the final population
• Fmat
  if algo$storeF is TRUE, a matrix of size algo$nG times algo$nP. Each column contains the best objective function value found by the particular solution.
• xlist
  if algo$storeSolutions is TRUE, a list that contains two lists P and Pbest of matrices; else NA.

code

max

Details

The function implements Particle Swarm Optimisation (PS); see the references for details on the implementation. PS is a population-based optimisation heuristic. It develops several solutions (a population) over a number of iterations. PS is directly applicable to continuous problems since the population is stored in real-valued vectors. In each iteration, a solution is updated by adding another vector called velocity. Think of a solution as a position in the search space, and of velocity as the direction into which this solution moves. Velocity changes over the course of the optimization: it is biased towards the best solution found by the particular solution and the best overall solution. The algorithm stops after a fixed number of iterations. To allow for constraints, the evaluation works as follows: after a new solution is created, it is (i) repaired, (ii) evaluated through the objective function, (iii) penalised. Step (ii) is done by a call to OF; steps (i) and (iii) by calls to algo$repair and algo$pen. Step (i) and (iii) are optional, so the respective functions default to NULL. A penalty can also be directly written in the OF, since it amounts to a positive number added to the clean objective function value. It can be advantageous to write a separate penalty function if either only the objective function or only the penalty function can be vectorised. (Constraints can also be added without these mechanisms. Solutions that violate constraints can, for instance, be mapped to feasible solutions, but without actually changing them. See Maringer and Oyewumi, 2007, for an example with Differential Evolution.) Conceptually, PS consists of two loops: one loop across the iterations and, in any given generation, one loop across the solutions. This is the default, controlled by the variables algo$loopOF, algo$loopRepair and algo$loopPen, which all default to TRUE. But it does not matter in what order the solutions are evaluated (or repaired or penalised), so the second loop can be vectorised. Examples are given in the vignettes and in the book. The respective algo$loopFun must then be set to FALSE. The objective function OF, the repair function and and the penalty function will be called as fun(solution, ...). The list algo contains the following items: [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

References

Eberhart, R.C. and Kennedy, J. (1995) A New Optimizer using Particle Swarm theory. Proceedings of the Sixth International Symposium on Micromachine and Human Science, pp. 39--43. Gilli, M., Maringer, D. and Schumann, E. (2011) Numerical Methods and Optimization in Finance. Elsevier. http://www.elsevierdirect.com/product.jsp?isbn=9780123756626 Schumann, E. (2011) Examples and Extensions for the NMOF Package. http://ssrn.com/abstract=1886522

See Also

DEopt

Examples

## Least Median of Squares (LMS) estimation
genData <- function(nP, nO, ol, dy) {
    ### create dataset as in Salibian-Barrera & Yohai 2006
    ### nP = regressors, nO  = number of obs
    ### ol = number of outliers, dy  = outlier size 
    mRN <- function(m, n) array(rnorm(m * n), dim = c(m, n))
    y <- mRN(nO, 1)
    X <- cbind(as.matrix(numeric(nO) + 1), mRN(nO, nP - 1L))
    zz <- sample(nO)
    z  <- cbind(1, 100, array(0, dim = c(1L, nP - 2L)))
    for (i in 1L:ol) {
        X[zz[i], ] <- z
        y[zz[i]] <- dy 
    }
    list(X = X, y = y)
}

OF <- function(param, data) {
    X <- data$X
    y <- data$y
    aux <- as.vector(y) - X %*% param
    ### as.vector(y) for recycling; param is a matrix
    aux <- aux * aux
    aux <- apply(aux, 2, sort, partial = data$h)
    aux[h, ]
}

nP <- 2L; nO <- 100L; ol <- 10L; dy <- 150
aux <- genData(nP,nO,ol,dy); X <- aux$X; y <- aux$y

h <- (nO + nP + 1L) %/% 2
data <- list(y = y,X = X, h = h)

algo <- list(min = rep(-10, nP), max = rep( 10, nP),
    c1 = 1.0, c2 = 2.0,
    iner = 0.7, initV = 1, maxV = 3, 
    nP = 100L, nG = 300L, loopOF = FALSE)

system.time(sol <- PSopt(OF = OF, algo = algo, data = data))
if (require("MASS", quietly = TRUE)) {
    ### for nsamp = "best", in this case, complete enumeration 
    ### will be tried. See ?lqs
    system.time(test1 <- lqs(data$y ~ data$X[ ,-1L], 
            adjust = TRUE, 
            nsamp = "best", 
            method = "lqs", 
            quantile = data$h))
}
## check
x1 <- sort((y - X %*% as.matrix(sol$xbest))^2)[h]
cat("Particle Swarm
",x1,"")

if (require("MASS", quietly = TRUE)) {
    x2 <- sort((y - X %*% as.matrix(coef(test1)))^2)[h]
    cat("lqs
",x2,"")
}