cigar: Bent Cigar Function

In gek: Gradient-Enhanced Kriging

Bent Cigar Function

Description

The Bent cigar function is defined by

f_{\rm cigar}(x_1, ..., x_d) = x_1^2 + 10^6 \sum_{k = 2}^{d} x_k^2

with x_k \in [-100, 100] for k = 1, ..., d.

Usage

cigar(x)
cigarGrad(x)

Arguments

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

Details

The gradient of the bent cigar function is

\nabla f_{\rm cigar}(x_1, ..., x_d) = \begin{pmatrix} 2x_1 \\ 20^6x_2\\ \vdots \\ 20^6x_d\end{pmatrix}.

The bent cigar function has one global minimum f_{\rm cigar}(x^{\star}) = 0 at x^{\star} = (1,\dots, 1).

Value

cigar returns the function value of the bent cigar function at x.

cigarGrad returns the gradient of the bent cigar 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")}.

See Also

tangents for drawing tangent lines.

Examples

# 1-dimensional Cigar function with tangents
curve(cigar(x), from = -5, to = 5, n = 200)
x <- seq(-4.5, 4.5, length = 5)
y <- cigar(x)
dy <- cigarGrad(x)
tangents(x, y, dy, length = 2, lwd = 2, col = "red")
points(x, y, pch = 16)

# Contour plot of Cigar function 
n.grid <- 50
x1 <- x2 <- seq(-100, 100, length.out = n.grid)
y <- outer(x1, x2, function(x1, x2) cigar(cbind(x1, x2)))
contour(x1, x2, y, xaxs = "i", yaxs = "i", nlevels = 25, xlab = "x1", ylab = "x2")

# Perspective plot of Cigar 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.