copulaedas (version 1.2.1)

problems: Benchmark Problems

Description

Implementation of a group of well-known benchmark problems usually used to evaluate the performance of EDAs and other 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.

Value

• The value of the function for the vector x.

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.

References

Bengoetxea E, Miqu�lez T, Lozano JA, Larra�aga P (2002). Experimental Results in Function Optimization with EDAs in Continuous Domain. In P Larra�aga, 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 Cant�-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.

Gonz�lez-Fern�ndez Y and Soto M (2012). copulaedas: An R Package for Estimation of Distribution Algorithms Based on Copulas. Preprint http://arxiv.org/abs/1209.5429{arXiv:1209.5429 [cs.NE]}.

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)