GPareto (version 1.1.8)

ZDT1: Test functions of x

Description

Multi-objective test functions.

Usage

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)
OKA1(x)

Value

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

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.

• OKA1: [0,1]^2, initially [6 sin(pi/12), 6 sin(pi/12) + 2pi cos(pi/12)] x [-2pi sin(pi/12), 6 cos(pi/12)], bi-objective

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.

T. Okabe, J. Yaochu, M. Olhofer, B. Sendhoff (2004), On test functions for evolutionary multi-objective optimization, International Conference on Parallel Problem Solving from Nature, Springer, Berlin, Heidelberg.

Examples

# ----------------------------------
# 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)