problems: Benchmark Problems
In yasserglez/copulaedas: Estimation of Distribution Algorithms Based on Copulas
Description
Usage
Arguments
Details
Value
References
Examples
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
1
2
3
4
5
6
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},…,x_{n}) is given below.
\texttt{fAckley}(\boldsymbol{x})=-20\exp≤ft(-0.2√{\frac{1}{n}∑_{i=1}^{n}x^{2}}\right)-\exp≤ft(\frac{1}{n}∑_{i=1}^{n}\cos≤ft(2π x_{i}\right)\right)+20+\exp≤ft(1\right)
\texttt{fGriewank}(\boldsymbol{x})=1+∑_{i=1}^{n}\frac{x_{i}^{2}}{4000}-∏_{i=1}^{n}\cos≤ft(\frac{x_{i}}{√{i}}\right)
\texttt{fRastrigin}(\boldsymbol{x})=∑_{i=1}^{n}≤ft(x_i^2 - 10 \cos≤ft(2 π x_i\right) + 10\right)
\texttt{fRosenbrock}(\boldsymbol{x})=∑_{i=1}^{n-1}≤ft(100≤ft(x_{i+1}-x_{i}^{2}\right)^{2}+≤ft(1-x_{i}\right)^{2}\right)
\texttt{fSphere}(\boldsymbol{x})=∑_{i=1}^{n}x_{i}^{2}
\texttt{fSummationCancellation}(\boldsymbol{x})=\frac{-1}{10^{-5}+∑_{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,…,0)
with evaluation 0. Rosenbrock has its global optimum at
\boldsymbol{x}=(1,…,1) with evaluation 0. Summation Cancellation
has its global optimum at \boldsymbol{x}=(0,…,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
1
2
3
4
5
6
yasserglez/copulaedas documentation built on June 9, 2021, 10:05 a.m.
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Description Usage Arguments Details Value References Examples
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
1 2 3 4 5 6 | 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},…,x_{n}) is given below.
\texttt{fAckley}(\boldsymbol{x})=-20\exp≤ft(-0.2√{\frac{1}{n}∑_{i=1}^{n}x^{2}}\right)-\exp≤ft(\frac{1}{n}∑_{i=1}^{n}\cos≤ft(2π x_{i}\right)\right)+20+\exp≤ft(1\right)
\texttt{fGriewank}(\boldsymbol{x})=1+∑_{i=1}^{n}\frac{x_{i}^{2}}{4000}-∏_{i=1}^{n}\cos≤ft(\frac{x_{i}}{√{i}}\right)
\texttt{fRastrigin}(\boldsymbol{x})=∑_{i=1}^{n}≤ft(x_i^2 - 10 \cos≤ft(2 π x_i\right) + 10\right)
\texttt{fRosenbrock}(\boldsymbol{x})=∑_{i=1}^{n-1}≤ft(100≤ft(x_{i+1}-x_{i}^{2}\right)^{2}+≤ft(1-x_{i}\right)^{2}\right)
\texttt{fSphere}(\boldsymbol{x})=∑_{i=1}^{n}x_{i}^{2}
\texttt{fSummationCancellation}(\boldsymbol{x})=\frac{-1}{10^{-5}+∑_{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,…,0) with evaluation 0. Rosenbrock has its global optimum at \boldsymbol{x}=(1,…,1) with evaluation 0. Summation Cancellation has its global optimum at \boldsymbol{x}=(0,…,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
1 2 3 4 5 6 |
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