makeDTLZ5Function function

DTLZ5 Function (family)

Builds and returns the multi-objective DTLZ5 test problem. This problem can be characterized by a disconnected Pareto-optimal front in the search space. This introduces a new challenge to evolutionary multi-objective optimizers, i.e., to maintain different subpopulations within the search space to cover the entire Pareto-optimal front.

The DTLZ5 test problem is defined as follows:

Minimize c("$\n$", "$f[1](X) = (1 + g(XM)) * cos(theta[1] * pi/2) * cos(theta[2] * pi/2) * ... * cos(theta[M-2] * pi/2) * cos(theta[M-1] * pi/2)$")

Minimize c("$\n$", "$f[2](X) = (1 + g(XM)) * cos(theta[1] * pi/2) * cos(theta[2] * pi/2) * ... * cos(theta[M-2] * pi/2) * sin(theta[M-1] * pi/2)$")

Minimize c("$\n$", "$f[3](X) = (1 + g(XM)) * cos(theta[1] * pi/2) * cos(theta[2] * pi/2) * ... * sin(theta[M-2] * pi/2)$")

$...$

Minimize c("$\n$", "$f[M-1](X) = (1 + g(XM)) * cos(theta[1] * pi/2) * sin(theta[2] * pi/2)$")

Minimize c("$\n$", "$f[M](X) = (1 + g(XM)) * sin(theta[1] * pi/2)$")

with $0 <= x[i] <= 1$, for $i=1,2,...,n$

where c("$\n$", "$theta[i] = pi / (4 * (1 + g(XM))) * (1 + 2 * g(XM) * x[i]),$")

for $i = 2,3,...,(M-1)$

and c("$\n$", "$g(XM) = sum{x[i] in XM} {(x[i] - 0.5)^2}$")

makeDTLZ5Function(dimensions, n.objectives)

Arguments

• dimensions: [integer(1)]

  Number of decision variables.

• n.objectives: [integer(1)]

  Number of objectives.

Returns

[smoof_multi_objective_function]

Note

This problem definition does not exist in the succeeding work of Deb et al. (K. Deb and L. Thiele and M. Laumanns and E. Zitzler (2002). Scalable multi-objective optimization test problems, Proceedings of the IEEE Congress on Evolutionary Computation, pp. 825-830).

Also, note that in case of a bi-objective scenario (n.objectives = 2L) DTLZ2 and DTLZ5 are identical.

References

K. Deb and L. Thiele and M. Laumanns and E. Zitzler. Scalable Multi-Objective Optimization Test Problems. Computer Engineering and Networks Laboratory (TIK), Swiss Federal Institute of Technology (ETH) Zurich, 112, 2001