schwefel: Schwefel Function

In gek: Gradient-Enhanced Kriging

Schwefel Function

Description

The Schwefel function is defined by

f_{\rm schwefel}(x_1, ..., x_d) = 418.9829 d - \sum_{k = 1}^{d} x_k \sin\left(\sqrt{|x_k|}\right)

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

Usage

schwefel(x)
schwefelGrad(x)

Arguments

x	a numeric vector of length d or a numeric matrix with n rows and d columns.

Details

The gradient of the Schwefel function is

\nabla f_{\rm schwefel}(x_1, \dots, x_d) = \begin{pmatrix} -\sin\left( \sqrt{|x_1|} \right) - \frac{x_1^2 \cos\left(\sqrt{|x_1|}\right)}{2 |x_1|^{\frac{3}{2}}} \\ \vdots \\ -\sin\left( \sqrt{|x_d|} \right) - \frac{x_d^2 \cos\left(\sqrt{|x_d|}\right)}{2 |x_d|^{\frac{3}{2}}} \end{pmatrix}.

The Schwefel function has one global minimum f_{\rm schwefel}(x^{\star}) = 0 at x^{\star} = (420.968746, \dots, 420.968746).

Value

schwefel returns the function value of the Schwefel function at x.

schwefelGrad returns the gradient of the Schwefel function at x.

Author(s)

Carmen van Meegen

References

Plevris, V. and Solorzano, G. (2022). A Collection of 30 Multidimensional Functions for Global Optimization Benchmarking. Data, 7(4):46. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.3390/data7040046")}.

Surjanovic, S. and Bingham, D. (2013). Virtual Library of Simulation Experiments: Test Functions and Datasets. https://www.sfu.ca/~ssurjano/ (retrieved January 19, 2024).

See Also

tangents for drawing tangent lines.

Examples

# 1-dimensional Schwefel function with tangents
curve(schwefel(x), from = -500, to = 500, n = 500)
x <- seq(-450, 450, length = 5)
y <- schwefel(x)
dy <- schwefelGrad(x)
tangents(x, y, dy, length = 200, lwd = 2, col = "red")
points(x, y, pch = 16)

# Contour plot of Schwefel function 
n.grid <- 75
x1 <- x2 <- seq(-500, 500, length.out = n.grid)
y <- outer(x1, x2, function(x1, x2) schwefel(cbind(x1, x2)))
contour(x1, x2, y, xaxs = "i", yaxs = "i", nlevels = 25, xlab = "x1", ylab = "x2")

# Perspective plot of Schwefel 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.