rastrigin: Rastrigin Function

In gek: Gradient-Enhanced Kriging

Rastrigin Function

Description

Rastrigin function is defined by

f_{\rm rastrigin}(x_1, ..., x_d) = 10 d + \sum_{k = 1}^{d} (x_k^2 - 10\cos(2\pi x_k))

with x_k \in [-5.12, 5.12] for k = 1, ..., d.

Usage

rastrigin(x)
rastriginGrad(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 Rastrigin function is

\nabla f_{\rm rastrigin}(x_1, ..., x_d) = \begin{pmatrix} 2x_1 + 20 \sin(2\pi x_1) \\ \vdots \\ 2x_d + 20 \sin(2\pi x_d))\end{pmatrix}.

Rastrigin function has one global minimum f_{\rm rastrigin}(x^{\star}) = 0 at x^{\star} = (0,\dots, 0).

Value

rastrigin returns the function value of Rastrigin function at x.

rastriginGrad returns the gradient of Rastrigin 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 Rastrigin function with tangents
curve(rastrigin(x), from = -5, to = 5, n = 200)
x <- seq(-4.5, 4.5, length = 5)
y <- rastrigin(x)
dy <- rastriginGrad(x)
tangents(x, y, dy, length = 2, lwd = 2, col = "red")
points(x, y, pch = 16)

# Contour plot of Rastrigin function 
n.grid <- 100
x1 <- x2 <- seq(-5.12, 5.12, length.out = n.grid)
y <- outer(x1, x2, function(x1, x2) rastrigin(cbind(x1, x2)))
contour(x1, x2, y, xaxs = "i", yaxs = "i", nlevels = 25, xlab = "x1", ylab = "x2")

# Perspective plot of Rastrigin 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.