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.
fAckley(x)
fGriewank(x)
fRosenbrock(x)
fRastrigin(x)
fSphere(x)
fSummationCancellation(x)A vector to be evaluated in the function.
The value of the function for the vector x.
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.
Bengoetxea E, Miqu<U+00E9>lez T, Lozano JA, Larra<U+00F1>aga P (2002). Experimental Results in Function Optimization with EDAs in Continuous Domain. In P Larra<U+00F1>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<U+00FA>-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/.
# NOT RUN {
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)
# }
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