problems {copulaedas}

Benchmark Problems

Description

Implementation of a group of well-known benchmark problems typically used to evaluate the performance of EDAs and other numerical optimization algorithms for unconstrained global optimization.

Usage

fAckley(x)
fGriewank(x)
fRosenbrock(x)
fRastrigin(x)
fSphere(x)
fSummationCancellation(x)

Arguments

x

A vector to be evaluated in the function.

Details

The definition of the functions for a vector \boldsymbol{x}=(x_{1},\ldots,x_{n}) is given below.

\texttt{fAckley}(\boldsymbol{x})=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x^{2}}\right)-\exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos\left(2\pi x_{i}\right)\right)+20+\exp\left(1\right)

\texttt{fGriewank}(\boldsymbol{x})=1+\sum_{i=1}^{n}\frac{x_{i}^{2}}{4000}-\prod_{i=1}^{n}\cos\left(\frac{x_{i}}{\sqrt{i}}\right)

\texttt{fRastrigin}(\boldsymbol{x})=\sum_{i=1}^{n}\left(x_i^2 - 10 \cos\left(2 \pi x_i\right) + 10\right)

\texttt{fRosenbrock}(\boldsymbol{x})=\sum_{i=1}^{n-1}\left(100\left(x_{i+1}-x_{i}^{2}\right)^{2}+\left(1-x_{i}\right)^{2}\right)

\texttt{fSphere}(\boldsymbol{x})=\sum_{i=1}^{n}x_{i}^{2}

\texttt{fSummationCancellation}(\boldsymbol{x})=\frac{-1}{10^{-5}+\sum_{i=1}^{n}|y_{i}|},\, y_{1}=x_{1},\, y_{i}=y_{i-1}+x_{i}

Ackley, Griewank, Rastrigin, Rosenbrock, and Sphere are minimization problems. Summation Cancellation is originally a maximization problem but it is expressed here as a minimization problem. Ackley, Griewank, Rastrigin and Sphere have their global optimum at \boldsymbol{x}=(0,\ldots,0) with evaluation 0. Rosenbrock has its global optimum at \boldsymbol{x}=(1,\ldots,1) with evaluation 0. Summation Cancellation has its global optimum at \boldsymbol{x}=(0,\ldots,0) with evaluation -10^{5}. See (Bengoetxea et al. 2002; Bosman and Thierens 2006; Chen and Lim 2008) for a description of the functions.

Value

The value of the function for the vector x.

References

Bengoetxea E, Miquélez T, Lozano JA, Larrañaga P (2002). Experimental Results in Function Optimization with EDAs in Continuous Domain. In P Larrañaga, JA Lozano (eds.), Estimation of Distribution Algorithms. A New Tool for Evolutionary Computation, pp. 181–194. Kluwer Academic Publisher

Bosman PAN, Thierens D (2006). Numerical Optimization with Real-Valued Estimation of Distribution Algorithms. In M Pelikan, K Sastry, E Cantú-Paz (eds.), Scalable Optimization via Probabilistic Modeling. From Algorithms to Applications, pp. 91–120. Springer-Verlag.

Chen Yp, Lim MH (eds.) (2008). Linkage in Evolutionary Computation. Springer-Verlag. ISBN 978-3-540-85067-0.

Gonzalez-Fernandez Y, Soto M (2014). copulaedas: An R Package for Estimation of Distribution Algorithms Based on Copulas. Journal of Statistical Software, 58(9), 1-34. http://www.jstatsoft.org/v58/i09/.

Examples

all.equal(fAckley(rep(0, 10)), 0)
all.equal(fGriewank(rep(0, 10)), 0)
all.equal(fRastrigin(rep(0, 10)), 0)
all.equal(fRosenbrock(rep(1, 10)), 0)
all.equal(fSphere(rep(0, 10)), 0)
all.equal(fSummationCancellation(rep(0, 10)), -1e+05)