qing: Qing Function

In gek: Gradient-Enhanced Kriging

Qing Function

Description

Qing function is defined by

f_{\rm qing}(x_1, ..., x_d) = \sum_{k = 1}^{d} (x_i^2 - i)^2

with x_k \in [-500, 500] for k = 1, ..., d.

Usage

qing(x)
qingGrad(x)

Arguments

x	a numeric vector or a numeric matrix with n rows and d columns. If a vector is passed, the 1-dimensional version of the Rastrigin function is calculated.

Details

The gradient of Qing function is

\nabla f_{\rm qing}(x_1, ..., x_d) = \begin{pmatrix} 4 x_1 (x_1^2 - 1) \\ \vdots \\ 4 x_d (x_d^2 - d) \end{pmatrix}.

Qing function has 2^d global minimum f_{\rm qing}(x^{\star}) = 0 at x^{\star} = (\pm \sqrt{1},\dots, \pm \sqrt{d}).

Value

qing returns the function value of Qing function at x.

qingGrad returns the gradient of Qing function at x.

Author(s)

Carmen van Meegen

References

Qing, A. (2006). Dynamic Differential Evolution Strategy and Applications in Electromagnetic Inverse Scattering Problems. IEEE Transactions on Geoscience and Remote Sensing, 44(1):116-–125. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1109/TGRS.2005.859347")}.

Jamil, M. and Yang, X.-S. (2013). A Literature Survey of Benchmark Functions for Global Optimization Problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2):150-–194. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1504/IJMMNO.2013.055204")}.

Examples

# 1-dimensional Qing function with tangents
curve(qing(x), from = -1.7, to = 1.7)
x <- seq(-1.5, 1.5, length = 5)
y <- qing(x)
dy <- qingGrad(x)
tangents(x, y, dy, length = 1, lwd = 2, col = "red")
points(x, y, pch = 16)

# Contour plot of Qing function 
n.grid <- 50
x1 <- seq(-2, 2, length.out = n.grid)
x2 <- seq(-2, 2, length.out = n.grid)
y <- outer(x1, x2, function(x1, x2) qing(cbind(x1, x2)))
contour(x1, x2, y, xaxs = "i", yaxs = "i", nlevels = 25, xlab = "x1", ylab = "x2")

# Perspective plot of Qing function
col.pal <- colorRampPalette(c("#00007F", "blue", "#007FFF", "cyan", "#7FFF7F", "yellow",
	"#FF7F00", "red", "#7F0000"))
colors <- col.pal(100)
y.facet.center <- (y[-1, -1] + y[-1, -n.grid] + y[-n.grid, -1] + y[-n.grid, -n.grid])/4
y.facet.range <- cut(y.facet.center, 100)
persp(x1, x2, y, phi = 30, theta = -315, expand = 0.75, ticktype = "detailed", 
	col = colors[y.facet.range])

gek documentation built on April 4, 2025, 12:35 a.m.