test_functions.Rd

%% Part of the hydroPSO R package, http://www.rforge.net/hydroPSO/ ; 
%%                                 http://cran.r-project.org/web/packages/hydroPSO
%% Copyright 2011-2012 Mauricio Zambrano-Bigiarini & Rodrigo Rojas
%% Distributed under GPL 2 or later

\name{test_functions}
\alias{ackley}
\alias{griewank}
\alias{rastrigrin}
\alias{rastrigin}
\alias{rosenbrock}
\alias{schafferF6}
\alias{schwefel}
\alias{sphere}
\alias{sackley}
\alias{sgriewank}
\alias{srastrigin}
\alias{srosenbrock}
\alias{sschwefel1_2}
\alias{ssphere}

%- Also NEED an '\alias' for EACH other topic documented here.
\title{
Test Functions for Global Optimisation
}
\description{
Test functions commonly used as benchmark for global optimisation problems
}
\usage{
ackley(x)
griewank(x)
rastrigrin(x)
rosenbrock(x)
schafferF6(x)
schwefel(x)
sphere(x)
sackley(x, o=-32+64*runif(length(x)), fbias=-140)
sgriewank(x, o=-600+1200*runif(length(x)), fbias=-180)
srastrigin(x, o=-5+10*runif(length(x)), fbias=-330)
srosenbrock(x, o=-100+200*runif(length(x)), fbias=390)
sschwefel1_2(x, o=-100+200*runif(length(x)), fbias=-450)
ssphere(x, o=-100+200*runif(length(x)), fbias=-450)

}
%- maybe also 'usage' for other objects documented here.
\arguments{
  \item{x}{
numeric vector to be evaluated
}
  \item{o}{
numeric shifting vector to be used, with the same length of \code{x}
}
  \item{fbias}{
numeric with the bias to be imposed
}
}
\details{
The \bold{Ackley} test function is multimodal and separable, with several local optima that, for the search range [-32, 32], look more like noise, although they are located at regular intervals. The Ackley function only has one global optimum located at the point \kbd{o=(0,...,0)}. It is defined by:
\deqn{ ackley = 20+\exp(1)-20\exp\left( -0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_{i}^2} \right)-\exp\left(\frac{1}{\textcolor{red}{n}}\sum_{i=1}^{n}\cos(2\pi x_{i})\right) ; -32 \leq x_i \leq 32 \ ; \ i=1,2,\ldots,n }


The generalized \bold{Rastrigin} test function is non-convex, multimodal and additively separable. It has several local optima arranged in a regular lattice, but it only has one global optimum located at the point \kbd{o=(0,...,0)}. The search range for the Rastrigin function is [-5.12, 5.12] in each variable. This function is a fairly difficult problem due to its large search space and its large number of local minima. It is defined by:

\deqn{ rastrigin = 10n+\sum_{i=1}^{n}\left[x_{i}^{2}-10\cos(2\pi x_{i})\right] \ ; \ -5.12 \leq x_i \leq 5.12 \ ; \ i=1,2,\ldots,n }


The \bold{Griewank} test function is multimodal and non-separable, with has several local optima within the search region defined by [-600, 600]. It is similar to the Rastrigin function, but the number of local optima is larger in this case. It only has one global optimum located at the point \kbd{o=(0,...,0)}. The function interpretation changes with the scale; the general overview suggests convex function, medium-scale view suggests existence of local extremum, and finally zoom on the details indicates complex structure of numerous local minima. While this function has an exponentially increasing number of local minima as its dimension increases, it turns out that a simple multistart algorithm is able to detect its global minimum more and more easily as the dimension increases (Locatelli, 2003). It is defined by:

\deqn{ griewank = \frac{1}{4000}\sum_{i=1}^{n}x_{i}^{2}-\prod_{i=1}^{n}\cos\left(\frac{x_i}{\sqrt{i}}\right)+1 \ ; \ -600 \leq x_i \leq 600 \ ; \ i=1,2,\ldots,n }


The \bold{Rosenbrock} function is non-convex, unimodal and non-separable. It is also known as \emph{Rosenbrock's valley} or \emph{Rosenbrock's banana} function. The global minimum is inside a long, narrow, parabolic shaped flat valley. To find the valley is trivial. To converge to the global minimum, however, is difficult. It only has one optimum located at the point \kbd{o=(1,...,1)}. It is a quadratic function, and its search range is [-30, 30] for each variable. It is defined by:

\deqn{ rosenbrock = \sum_{i=1}^{n-1}\left[100(x_{i+1}\textcolor{red}{-}x_{i}^{2})^{2}+(1-x_{i})^{2}\right] \ ; \ -30 \leq x_i \leq 30 \ ; \ i=1,2,\ldots,n }


The main difficulty of the \bold{Schaffer's F6} test function is that the size of the potential maxima that need to be overcomed to get to a minimum increases the closer
one gets to the global minimum. It is defined by:

\deqn{ schafferF6 = 0.5+\frac{\sin^{2}\sqrt{\textcolor{red}{\sum_{i=1}^{n}}x_{i}^{2}}-0.5}{(1+0.001\textcolor{red}{\sum_{i=1}^{n}}x_{i}^{2})^{2}} \ ; \ -100 \leq x_i \leq 100 \ ; \ i=1,2,\ldots,n } \cr


The \emph{first function of De Jong's} or \bold{Sphere} function is one of the most simple test functions available in the specialized literature. This continuous, convex, unimodal and additively separable test function can be scaled up to any number of variables. It belongs to a family of functions called quadratic functions and only has one optimum in the point \kbd{o=(0,...,0)}. The search range commonly used for the Sphere function is [-100, 100] for each decision variable. It is defined by:

\deqn{ sphere = \sum_{i=1}^{n} x_{i}^2 \ ; \ -100 \leq x_i \leq 100 \ ; \ i=1,2,\ldots,n  } \cr


The \bold{Schwefel's} function is non-convex, multimodal, and additively separable. It is deceptive in that the global minimum is geometrically distant, over the parameter space, from the next best local minima. Therefore, the search algorithms are potentially prone to convergence in the wrong direction. In addition, it is less symmetric than the Rastrigin function and has the global minimum at the edge of the search space [-500, 500] at position \kbd{o=(420.9687,...,420.9687)}. Additionally, there is no overall, guiding slope towards the global minimum like in Ackley's, or less extreme, in Rastrigin's function. It is defined by:

\deqn{ schwefel = 418.982887274338n + \sum_{i=1}^{n} -x_{i} \sin(\sqrt{|x_{i}|} \ ; \ -500 \leq x_i \leq 500 \ ; \ i=1,2,\ldots,n  }


The \bold{Shifted Schwefel's Problem 1.2} function is unimodal, non-separable, and scalable. It is defined by:

\deqn{ sschwefel1\_2 = \sum_{i=1}^{n} {  \left(\sum_{j=1}^{i} {x_{j}} \right)^2 } + f\_bias \ ; \ -500 \leq x_i \leq 500 \ ; \ i=1,2,\ldots,n  } \cr


Some optimisation algorithms take advantage of known properties of the benchmark functions, such as local optima lying along the coordinate axes, global optimum having the same values for many variables and so on. In order to avoid the previous shortcomings, shifting vector and a single bias is introduced for some benchmark functions, reported afterwards.

The \bold{Shifted Ackley} is defined by:
\deqn{ sackley = 20+\exp(1)-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}z_{i}^2}\right)-\exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi z_{i})\right) + f\_bias , z=x-o ; \ i=1,2,\ldots,n } \cr

The \bold{Shifted Griewank} is defined by:
\deqn{ sgriewank = \frac{1}{4000}\sum_{i=1}^{n}z_{i}^{2}-\prod_{i=1}^{n}\cos\left(\frac{z_i}{\sqrt{i}}\right)+1 + f\_bias \ , \ z=x-o ; \ i=1,2,\ldots,n } \cr

The \bold{Shifted Sphere} is defined by:
\deqn{ ssphere = \sum_{i=1}^{n} z_{i}^2 + f\_bias \ , \ z=x-o ; \ i=1,2,\ldots,n }{%
sphere = sum( x^2 ) }
}
\value{
Each test function returns a single numeric value corresponding to the function evaluated on the vector \code{x}
%%  If it is a LIST, use
%%  \item{comp1 }{Description of 'comp1'}
%%  \item{comp2 }{Description of 'comp2'}
%% ...
}
\references{

\cite{Dieterich, J.M. and B.Hartke. 2012. Empirical review of standard benchmark functions using evolutionary global optimization. Appl.Math. 3. 1552-1564, DOI:10.4236/am.2012.330215}\cr

\cite{Barrera, J., and C. Coello Coello. 2010, Test function generators for assessing the performance of PSO algorithms in multimodal optimization, in Handbook of Swarm Intelligence, vol. 8, edited by B. Panigrahi, Y. Shi, and M.-H. Lim, chap. Adaptation, Learning, and Optimization, pp. 89-117, Springer Berlin Heidelberg, doi:10.1007/978-3-642-17390-5 4}\cr

Problem Definitions and Evaluation Criteria for the CEC 2005 Special Session on Real-Parameter Optimization \cite{\url{www.lri.fr/~hansen/Tech-Report-May-30-05.pdf}}\cr

Test functions for optimization needs: \cite{\url{http://www.zsd.ict.pwr.wroc.pl/files/docs/functions.pdf}}\cr


\bold{Web pages}:

GEATbx: Example Functions (single and multi-objective functions). \cite{\url{http://www.geatbx.com/docu/fcnindex-01.html}}\cr

Benchmark Problems \cite{\url{http://www.cs.cmu.edu/afs/cs/project/jair/pub/volume24/ortizboyer05a-html/node6.html}}\cr

Test Functions for Unconstrained Global Optimization \cite{\url{http://www-optima.amp.i.kyoto-u.ac.jp/member/student/hedar/Hedar_files/TestGO_files/Page364.htm}}\cr

Rosenbrock: \cite{\url{http://www.it.lut.fi/ip/evo/functions/node5.html}},  \cite{\url{http://en.wikipedia.org/wiki/Rosenbrock_function}}\cr

Sphere: \cite{\url{http://www.it.lut.fi/ip/evo/functions/node2.html}}\cr

Rastrigin: \cite{\url{http://www.it.lut.fi/ip/evo/functions/node6.html}}, \cite{\url{http://en.wikipedia.org/wiki/Rastrigin_function}}\cr

Ackley: \cite{\url{http://www.it.lut.fi/ip/evo/functions/node14.html}}\cr

Griewank: \cite{Locatelli, M. 2003. A note on the griewank test function, Journal of Global Optimization, 25 (2), 169-174, doi:10.1023/A:1021956306041}\cr

Schaffer's F6 \cite{Xiaohong Qiu, Jun Liu. 2009. A Novel Adaptive PSO Algorithm on Schaffer's F6 Function. Hybrid Intelligent Systems, International Conference on, pp. 94-98, 2009 Ninth International Conference on Hybrid Intelligent Systems} \cr

Schwefel: \cite{\url{http://www.geatbx.com/docu/fcnindex-01.html\#P150_6749}}
}

\author{
Mauricio Zambrano-Bigiarini, \email{mzb.devel@gmail.com}
}
%%\note{
%%  ~~further notes~~
%%}

%% ~Make other sections like Warning with \section{Warning }{....} ~

\seealso{
\code{\link{hydroPSO}}
}
%%\examples{

%%}
% Add one or more standard keywords, see file 'KEYWORDS' in the
% R documentation directory.
\keyword{optim}
\keyword{math}% __ONLY ONE__ keyword per line