Augmented Tchebycheff function

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Description

The Augmented Tchebycheff function (KNOWLES, 2006) is a scalarizing function witch the advantages of having a non-linear term. That causes points on nonconvex regions of the Pareto front can bve minimizers of this function and, thus, nonsupported solutions can be obtained.

Usage

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Tchebycheff(y, s = 100, rho = 0.1)

Arguments

y

Numerical matrix or data.frame containing the responses (on each column) to be scalarized.

s

Numerical integer (default: 100) setting the number of partitions the vector lambda has.

rho

A small positive value (default: 0.1) setting the "strenght" of the non-linear term.

References

Knowles, J. (2006). ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems. IEEE Transactions on Evolutionary Computation, 10(1), 50-66.

Examples

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grid <- expand.grid(seq(0, 1, , 50),seq(0, 1, , 50))
 res <- t(apply(grid, 1, nowacki_beam))
 plot(nowacki_beam_tps$x, xlim=c(0,1), ylim=c(0,1))
 grid <- grid[which(as.logical(apply(res[,-(1:2)] < 0, 1, prod))),]
 res <- res[which(as.logical(apply(res[,-(1:2)] < 0, 1, prod))),1:2]
for (i in 1:10){
sres <- Tchebycheff(res[,1:2], s=100, rho=0.1)
points(grid[which.min(sres),], col='green')
}