branin: Branin-Hoo Function

In gek: Gradient-Enhanced Kriging

Branin-Hoo Function

Description

The Branin-Hoo function is defined by

f_{\rm branin}(x_1, x_2) = \left(x_2 - \frac{5.1}{4 \pi^2}x_1^2 + \frac{5}{\pi}x_1 - 6\right)^2 + 10 \left(1-\frac{1}{8\pi}\right)\cos(x_1) + 10

with x_1 \in [-5, 10] and x_2 \in [0, 15].

Usage

branin(x)
braninGrad(x)

Arguments

x	a numeric vector of length 2 or a numeric matrix with n rows and 2 columns.

Details

The gradient of the Branin-Hoo function is

\nabla f_{\rm branin}(x_1, x_2) = \begin{pmatrix} 2 \left(x_2 - \frac{5.1 x_1^2}{4 \pi^2} + \frac{5 x_1}{\pi} - 6\right) \left(-10.2 \frac{x_1}{4\pi^2} + \frac{5}{\pi}\right) - 10 \left(1 - \frac{1}{8\pi}\right) \sin(x_1) \\ 2 \left( x_2 - \frac{5.1 x_1^2}{4 \pi^2} + \frac{5 x_1}{\pi} - 6\right)\end{pmatrix}.

The Branin-Hoo function has three global minima f_{\rm branin}(x^{\star}) = 0.397887 at x^{\star} = (-\pi, 12.275), x^{\star} = (\pi, 2.275) and x^{\star} = (9.42478, 2.475).

Value

branin returns the function value of the Branin-Hoo function at x.

braninGrad returns the gradient of the Branin-Hoo function at x.

Author(s)

Carmen van Meegen

References

Branin, Jr., F. H. (1972). Widely Convergent Method of Finding Multiple Solutions of Simultaneous Nonlinear Equations. IBM Journal of Research and Development, 16(5):504–522.

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

Examples

# Contour plot of Branin-Hoo function 
n.grid <- 50
x1 <- seq(-5, 10, length.out = n.grid)
x2 <- seq(0, 15, length.out = n.grid)
y <- outer(x1, x2, function(x1, x2) branin(cbind(x1, x2)))
contour(x1, x2, y, xaxs = "i", yaxs = "i", nlevels = 25, xlab = "x1", ylab = "x2")

# Perspective plot of Branin-Hoo 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.