makeDTLZ3Function function

DTLZ3 Function (family)

Builds and returns the multi-objective DTLZ3 test problem. The formula is very similar to the formula of DTLZ2, but it uses the $g$ function of DTLZ1, which introduces a lot of local Pareto-optimal fronts. Thus, this problems is well suited to check the ability of an optimizer to converge to the global Pareto-optimal front.

The DTLZ3 test problem is defined as follows:

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

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

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

$...$

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

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

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

where c("$\n$", "$g(XM) = 100 * (|XM| + sum{x[i] in XM} {(x[i] - 0.5)^2 - cos(20 * pi * (x[i] - 0.5))})$")

makeDTLZ3Function(dimensions, n.objectives)

Arguments

• dimensions: [integer(1)]

  Number of decision variables.

• n.objectives: [integer(1)]

  Number of objectives.

Returns

[smoof_multi_objective_function]

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