Cross-Entropy optimizer

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Description

CEopt is an optimization function based on the Cross-Entropy method

Usage

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	CEoptim(f, f.arg=NULL, maximize=FALSE, continuous=NULL, discrete=NULL,
	    N=100L, rho=0.1, iterThr=1e4L, noImproveThr=5, verbose=FALSE)

Arguments

f

Function to be optimized. Can have continuous and discrete arguments

f.arg

List of additional fixed arguments passed to function f.

maximize

Logical value determining whether to maximize or minimize the objective function

continuous

List of arguments for the continuous optimization part consisting of:

  • mean Vector of initial means.

  • sd Vector of initial standard deviations.

  • smoothMean Smoothing parameter for the vector of means. Default value 1 (no smoothing).

  • smoothSd Smoothing parameter for the standard deviations. Default value 1 (no smoothing).

  • sdThr Positive numeric convergence threshold. Check whether the maximum standard deviation is smaller than sdThr. Default value 0.001.

  • conMat Coefficient matrix of linear constraint conMat x ≤ conVec.

  • conVec Value vector of linear constraint conMat x ≤ conVec.

discrete

List of arguments for the discrete optimization part, consisting of:

  • categories Integer vector which defines the allowed values of the categorical variables. The ith categorical variable takes values in the set {0,1,...,categories(i)-1}.

  • probs List of initial probabilities for the categorical variables. Defaults to equal (uniform) probabilities.

  • smoothProb Smoothing parameter for the probabilities of the categorical sampling distribution. Default value 1 (no smoothing).

  • ProbThr Positive numeric convergence threshold. Check whether all probabilities in the categorical sampling distributions deviate less than ProbThr from either 0 or 1. Default value 0.001.

N

Integer representing the CE sample size.

rho

Value between 0 and 1 representing the elite proportion.

iterThr

Termination threshold on the largest number of iterations.

noImproveThr

Termination threshold on the largest number of iterations during which no improvement of the best function value is found.

verbose

Logical value set for CE progress output.

Value

CEoptim returns an object of class "CEoptim" which is a list with the following components.

  • optimum Optimal value of f.

  • optimizer List of the location of the optimal value, consisting of:

    • continuous Continuous part of the optimizer.

    • discrete Discrete part of the optimizer.

  • termination List of termination information consisting of:

    • niter Total number of iterations upon termination.

    • convergence One of the following statements:

      • Not converged, if the number of iterations reaches iterThr;

      • The optimum did not change for noImproveThr iterations, if the best value has not improved for noImproveThr iterations;

      • Variances converged, otherwise.

  • states List of intermediate results computed at each iteration. It consists of the iteration number (iter), the best overall value (optimum) and the worst value of the elite samples, (gammat). The means (mean) and maximum standard deviations (maxSd) of the elite set are also included for continuous cases, and the maximum deviations (maxProbs) of the sampling probabilities to either 0 or 1 are included for discrete cases.

  • states.probs List of categorical sampling probabilities computed at each iteration. Will only be returned for discrete and mixed cases.

Note

Although partial parameter passing is allowed outside lists, it is recommended that parameters names are specified in full. Parameters inside lists have to specified completely.

Because CEoptim is a random function it is useful to (1) set the seed for the random number generator (for testing purposes), and (2) investigate the quality of the results by repeating the optimization a number of times.

Author(s)

Tim Benham, Qibin Duan, Dirk P. Kroese, Benoit Liquet

References

Benham T., Duan Q., Kroese D.P., Liquet B. (2015) CEoptim: Cross-Entropy R package for optimization. Journal of Statistical Software, submitted.

Rubinstein R.Y. and Kroese D.P. (2004). The Cross-Entropy Method. Springer, New York.

Examples

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## Maximizing the Peaks Function


fun <- function(x){
return(3*(1-x[1])^2*exp(-x[1]^2 - (x[2]+1)^2)
	-10*(x[1]/5-x[1]^3 - x[2]^5)*exp(-x[1]^2 - x[2]^2)
	-1/3*exp(-(x[1]+1)^2 - x[2]^2))}

set.seed(1234)

mu0 <- c(-3,-3); sigma0 <- c(10,10)
 
res <- CEoptim(fun,continuous=list(mean=mu0, sd=sigma0), maximize=TRUE)

## To extract the Optimal value of fun
res$optimum
## To extract the location of the optimal value
res$optimizer$continuous
## print function gives the following default values
print(res)