griewank: Griewank Function

In gek: Gradient-Enhanced Kriging

Griewank Function

Description

Griewank function is defined by

f_{\rm griewank}(x_1, ..., x_d) = \sum_{k = 1}^{d} \frac{x_k^2}{4000} - \prod_{k = 1}^d \cos\left(\frac{x_k}{\sqrt{k}}\right) + 1

with x_k \in [-600, 600] for k = 1, ..., d.

Usage

griewank(x)
griewankGrad(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 Griewank function is calculated.

Details

The gradient of Griewank function is

\nabla f_{\rm griewank}(x_1, ..., x_d) = \begin{pmatrix} \frac{x_1}{2000} + \frac{1}{\sqrt{1}} \sin\left( \frac{x_1}{\sqrt{1}}\right) \prod_{k = 2}^d \cos\left(\frac{x_k}{\sqrt{k}}\right) \\ \vdots \\ \frac{x_d}{2000} + \frac{1}{\sqrt{d}} \sin\left( \frac{x_d}{\sqrt{d}}\right) \prod_{k = 1}^{d-1} \cos\left(\frac{x_k}{\sqrt{k}}\right) \end{pmatrix}.

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

Value

griewank returns the function value of Griewank function at x.

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

# Contour plot of Griewank function 
n.grid <- 50
x1 <- x2 <- seq(-5, 5, length.out = n.grid)
y <- outer(x1, x2, function(x1, x2) griewank(cbind(x1, x2)))
contour(x1, x2, y, xaxs = "i", yaxs = "i", nlevels = 25, xlab = "x1", ylab = "x2")

# Perspective plot of Griewank 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.