# TestFunctions: Test functions of x In GPareto: Gaussian Processes for Pareto Front Estimation and Optimization

## Description

Multi-objective test functions.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25``` ```ZDT1(x) ZDT2(x) ZDT3(x) ZDT4(x) ZDT6(x) P1(x) P2(x) MOP2(x) MOP3(x) DTLZ1(x, nobj = 3) DTLZ2(x, nobj = 3) DTLZ3(x, nobj = 3) DTLZ7(x, nobj = 3) ```

## Arguments

 `x` matrix specifying the location where the function is to be evaluated, one point per row, `nobj` optional argument to select the number of objective for the DTLZ test functions.

## Details

These functions are coming from different benchmarks: the `ZDT` test problems from an article of E. Zitzler et al., `P1` from the thesis of J. Parr and `P2` from an article of Poloni et al. . `MOP2` and `MOP3` are from Van Veldhuizen and `DTLZ` functions are from Deb et al. .

Domains (sometimes rescaled to `[0,1]`):

• `ZDT1-6`: `[0,1]^d`

• `P1`, `P2`: `[0,1]^2`

• `MOP2`: `[0,1]^d`

• `MOP3`: `[-3,3]`, tri-objective, 2 variables

• `DTLZ1-3,7`: `[0,1]^d`, m-objective problems, with at least `d>m` variables.

## Value

Matrix of values corresponding to the objective functions, the number of colums is the number of objectives.

## References

J. M. Parr (2012), Improvement Criteria for Constraint Handling and Multiobjective Optimization, University of Southampton, PhD thesis.

C. Poloni, A. Giurgevich, L. Onesti, V. Pediroda (2000), Hybridization of a multi-objective genetic algorithm, a neural network and a classical optimizer for a complex design problem in fluid dynamics, Computer Methods in Applied Mechanics and Engineering, 186(2), 403-420.

E. Zitzler, K. Deb, and L. Thiele (2000), Comparison of multiobjective evolutionary algorithms: Empirical results, Evol. Comput., 8(2), 173-195.

K. Deb, L. Thiele, M. Laumanns and E. Zitzler (2002), Scalable Test Problems for Evolutionary Multiobjective Optimization, IEEE Transactions on Evolutionary Computation, 6(2), 182-197.

D. A. Van Veldhuizen, G. B. Lamont (1999), Multiobjective evolutionary algorithm test suites, In Proceedings of the 1999 ACM symposium on Applied computing, 351-357.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19``` ```# ---------------------------------- # 2-objectives test problems # ---------------------------------- plotParetoGrid("ZDT1", n.grid = 21) plotParetoGrid("ZDT2", n.grid = 21) plotParetoGrid("ZDT3", n.grid = 21) plotParetoGrid("ZDT4", n.grid = 21) plotParetoGrid("ZDT6", n.grid = 21) plotParetoGrid("P1", n.grid = 21) plotParetoGrid("P2", n.grid = 21) plotParetoGrid("MOP2", n.grid = 21) ```

GPareto documentation built on May 31, 2021, 5:09 p.m.