Cross-Entropy R package for optimization

Description

The CEoptim package provides an optimization solver based on the Cross-Entropy method. The main function CEoptim can be used to solve multi-extremal optimization problems involving discrete, continuous, and mixed variables. In addition, CEoptim implements linear constraints for continuous optimization.

Author(s)

Tim Benham, Qibin Duan, Dirk P. Kroese, Benoit Liquet <b.liquet@uq.edu.au>

References

Benham T., Duan Q., Kroese D.P., Liquet B. (2017) CEoptim: Cross-Entropy R package for optimization. Journal of Statistical Software, 76(8), 1-29.

See Also

CEoptim

Cross-Entropy optimizer

Description

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

Usage

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

Arguments

• 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 \le conVec.

• conVec Value vector of linear constraint conMat x \le conVec.

• 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.

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. (2017) CEoptim: Cross-Entropy R package for optimization. Journal of Statistical Software, 76(8), 1-29.

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

Examples

## 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)

Simulated data from FitzHugh-Nagumo differential equations

Description

The data correspond to the values V(t) of the FitzHugh-Nagumo differential equations

V'(t) = c*(V(t) - (V(t)^3)/3 + R(t))

R'(t) = -(1/c)*(V(t) - a + b*R(t))

at times 0, 0.05,..,20.0, with parameters a = 0.2, b = 0.2, c = 3 and initial conditions V(0) = -1, R(0)=1, and adding gaussian noise with standard deviation 0.5.

Usage

data(FitzHugh)

Format

A numeric vetor of length 401

References

Nagumo, J. and Arimoto, S. and Yoshizawa, S. (1962) An active pulse ransmission line simulating nerve axon, Proceedings of the IRE, 50 (10), 2061–2070.

Ramsay, J.O. and Hooker, G. and Campbell, D. and Cao J. (2007) Parameter estimation for differential equations: A generalized smoothing approach, Journal of the Royal Statistical Society, Series B 69 (5) 741–796.

Benham T., Duan Q., Kroese D.P., Liquet B. (2017) CEoptim: Cross-Entropy R package for optimization. Journal of Statistical Software, 76(8), 1-29.

Examples

## Plot the data
data(FitzHugh)
plot(FitzHugh,col="blue")

Dirichlet generator

Description

Random generation for the Dirichlet distribution

Usage

  dirichletrnd(a, n) 

Arguments

Value

dirichletrnd generates n random observations from a Dirichlet distribution

Author(s)

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

References

Kroese D.P., Taimre T., Botev Z.I. (2011), Handbook of Monte Carlo Methods, John Wiley & Sons.

Examples

## Generation from the Dirichlet distribution 
## with parameter a=(1,2,3,4,5)

set.seed(12345)
a <- 1:5
n <- 10

y <- dirichletrnd(a,n)
y

Network data from Les Miserables

Description

An R implementation of Donald Knuth's social network graph describing the interaction of characters in Vicor Hugo's novel Les Miserables. Each node represents a character, and edges connect any pair of characters that coappear. The weights of the edges are the number of such coapperances.

Usage

data(lesmis)

Format

Matrix of weights (77x77)

References

Knuth, D.E. (1993) The Stanford GraphBase: A Platform for Combinatorial Computing, ACM Press: Reading MA

Benham T., Duan Q., Kroese D.P., Liquet B. (2017) CEoptim: Cross-Entropy R package for optimization. Journal of Statistical Software, 76(8), 1-29.

Examples

## Display the social network graph
data(lesmis)
gplot(lesmis,gmode="graph")

Print method for the CEoptim object

Description

Produce print method for class "CEoptim"

Usage

## S3 method for class 'CEoptim'
print(x,...)

Arguments

Details

print method for "CEoptim" class, returns by default the main description of the x object including: optimizer; optimum; termination. To get the states and states.probs outputs, one should specify the corresponding argument to "TRUE".

Author(s)

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

References

Benham T., Duan Q., Kroese D.P., Liquet B. (2017) CEoptim: Cross-Entropy R package for optimization. Journal of Statistical Software, 76(8), 1-29.

See Also

CEoptim

Examples

## 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)

## Print method provides by default 
## optimizer; optimum and termination.
print(res)
## To print only the Optimal value of fun
print(res,optimum=TRUE)
## To print only the location of the optimal value
print(res,optimizer=TRUE)
## To print only termination information
print(res,termination=TRUE)

Internal function used for generating truncated multivariate normal data

Description

Internal function used for generating truncated multivariate normal data

Usage

  rtmvnorm (N, mu, sigma, A, b,..., rhoThr=NULL, maxSample=NULL) 

Simulated cumulative data from an AR(1) model with regime switching

Description

yt represents the added value of a stock at time t, at day t=1,2,...,300; that is, the increase (which may be negative) in stock price relative to the price at time t=0.

Usage

data(yt)

Format

Numeric vector of length 300

References

Benham T., Duan Q., Kroese D.P., Liquet B. (2017) CEoptim: Cross-Entropy R package for optimization. Journal of Statistical Software, 76(8), 1-29.

Examples

## Plot the yt data
data(yt)
plot(yt,type="l",col="blue")