gek (version 1.2.0)

qing: 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)

Value

qing returns the function value of Qing function at x.

qingGrad returns the gradient of Qing function at 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.

Author

Carmen van Meegen

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 \(x^{\star} = (\pm \sqrt{1},\dots, \pm \sqrt{d})\).

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. 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. tools:::Rd_expr_doi("10.1504/IJMMNO.2013.055204").

See Also

testfunctions for further test functions.

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])