camel6: Six-Hump Camel Function
View source: R/testfunctions.R
| camel6 | R Documentation |
Six-Hump Camel Function
Description
The six-hump camel function is defined by
f_{\rm camel6}(x_1, x_2) = \left(4 - 2.1 x_1^2 + \frac{x_1^4}{3}\right) x_1^2 + x_1 x_2 + (-4 + 4 x_2^2) x_2^2
with x_1 \in [-3, 3] and x_2 \in [-2, 2].
Usage
camel6(x)
camel6Grad(x)
Arguments
x |
a numeric |
Details
The gradient of the six-hump camel function is
\nabla f_{\rm camel6}(x_1, x_2) = \begin{pmatrix} 8 x_1 - 8.4 x_1^3 + 2 x_1^5 + x_2 \\ x_1 - 8 x_2 + 16 x_2^3 \end{pmatrix}.
The six-hump camel function has two global minima f_{\rm camel6}(x^{\star}) = -1.031628 at x^{\star} = (0.0898, -0.7126) and x^{\star} = (-0.0898, 0.7126).
Value
camel6 returns the function value of the six-hump camel function function at x.
camel6Grad returns the gradient of the six-hump camel function function at x.
Author(s)
Carmen van Meegen
References
Jamil, M. and Yang, X.-S. (2013). A Literature Survey of Benchmark Functions for Global Optimization Problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2):150-–194. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1504/IJMMNO.2013.055204")}.
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 six-hump camel function
n.grid <- 50
x1 <- seq(-2, 2, length.out = n.grid)
x2 <- seq(-1, 1, length.out = n.grid)
y <- outer(x1, x2, function(x1, x2) camel6(cbind(x1, x2)))
contour(x1, x2, y, xaxs = "i", yaxs = "i", nlevels = 25, xlab = "x1", ylab = "x2")
# Perspective plot of six-hump camel 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])
