sphere: Sphere Function

In gek: Gradient-Enhanced Kriging

Sphere Function

Description

The sphere function is defined by

f_{\rm sphere}(x_1, ..., x_d) = \sum_{k = 1}^{d} x_k^2

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

Usage

sphere(x)
sphereGrad(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 sphere function is calculated.

Details

The gradient of the sphere function is

\nabla f_{\rm sphere}(x_1, \dots, x_d) = \begin{pmatrix} 2 x_1 \\ \vdots \\ 2 x_d \end{pmatrix}.

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

Value

sphere returns the function value of the sphere function at x.

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

# Contour plot of sphere function 
n.grid <- 50
x1 <- x2 <- seq(-5.12, 5.12, length.out = n.grid)
y <- outer(x1, x2, function(x1, x2) sphere(cbind(x1, x2)))
contour(x1, x2, y, xaxs = "i", yaxs = "i", nlevels = 25, xlab = "x1", ylab = "x2")

# Perspective plot of sphere 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.