ZDT1 | R Documentation |
Multi-objective test functions.
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)
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. |
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
Matrix of values corresponding to the objective functions, the number of colums is the number of objectives.
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.
# ---------------------------------- # 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)
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