makeED2Function function

ED2 Function

Builds and returns the multi-objective ED2 test problem.

The ED2 test problem is defined as follows:

Minimize c("$\n$", "$f[j](X) = (1 / (F[natmin](X) + 1)) * p(\\Theta(X))$"), for $j = 1, ..., m$,

with $X = (x[1], ..., x[n])$, where $0 \le x[i] \le 1$, and $\Theta = (\theta[1], ..., \theta[m-1])$, where $0 \le \theta[j] \le \pi/2$, for $i = 1, ..., n$ and $j = 1, ..., m - 1$.

Moreover $F[natmin](X) = b + (r(X) - a) + 0.5 + 0.5 * (2 \pi * (r(X) - a) + \pi)$

with $a = 0.051373$, $b = 0.0253235$, and $r(X) = sqrt(x[m]^2 + ... + x[n]^2)$, as well as

$p[1](\Theta) = cos(\theta[1])^(2/\gamma)$,

$p[j](\Theta) = (sin(\theta[1]) * ... * sin(\theta[j - 1]) * cos(\theta[j]))^(2/\gamma)$, for $2 \le j \le m - 1$,

and $p[m](\Theta) = (sin(\theta[1]) * ... * sin(\theta[m - 1]))^(2/\gamma)$.

makeED2Function(dimensions, n.objectives, gamma = 2, theta)

Arguments

• dimensions: [integer(1)]

  Number of decision variables.

• n.objectives: [integer(1)]

  Number of objectives.

• gamma: [numeric(1)]

  Optional parameter. Default is 2, which is recommended by Emmerich and Deutz.

• theta: [numeric(dimensions)]

  Parameter vector, whose components have to be between 0 and 0.5*pi. The default is theta = (pi/2) * x (with x being the point from the decision space) as recommended by Emmerich and Deutz.

Returns

[smoof_multi_objective_function]

References

M. T. M. Emmerich and A. H. Deutz. Test Problems based on Lame Superspheres. Proceedings of the International Conference on Evolutionary Multi-Criterion Optimization (EMO 2007), pp. 922-936, Springer, 2007.