# makeDTLZ3Function: DTLZ3 Function (family) In smoof: Single and Multi-Objective Optimization Test Functions

## Description

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 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 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 f[3](X) = (1 + g(XM)) * cos(x[1] * pi/2) * cos(x[2] * pi/2) * ... * sin(x[M-2] * pi/2)

...

Minimize f[M-1](X) = (1 + g(XM)) * cos(x[1] * pi/2) * sin(x[2] * pi/2)

Minimize f[M](X) = (1 + g(XM)) * sin(x[1] * pi/2)

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

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

## Usage

 `1` ```makeDTLZ3Function(dimensions, n.objectives) ```

## Arguments

 `dimensions` [`integer(1)`] Number of decision variables. `n.objectives` [`integer(1)`] Number of objectives.

## Value

[`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

smoof documentation built on Aug. 14, 2017, 5:02 p.m.