banana: Rosenbrock's Banana Function

In gek: Gradient-Enhanced Kriging

Rosenbrock's Banana Function

Description

Rosenbrock's banana function is defined by

f_{\rm banana}(x_1, ..., x_d) = \sum_{k = 1}^{d - 1} (100 (x_{k+1} - x_k^2)^2 + (x_k - 1)^2)

with x_k \in [-5, 10] for k = 1, ..., d and d \geq 2.

Usage

banana(x)
bananaGrad(x)

Arguments

x	a numeric vector of length d or a numeric matrix with n rows and d columns, where d must be greater than 1.

Details

The gradient of Rosenbrock's banana function is

\nabla f_{\rm banana}(x_1, ..., x_d) = \begin{pmatrix} -400 (x_2 - x_1)^2 x_1 + 2 (x_1 - 1) \\ 200 (x_2 - x_1)^2 - 400 x_2 (x_3 - x_2^2) + 2 (x_2 - 1) \\ \vdots \\ 200 (x_{d-1} - x_{d-2})^2 - 400 x_{d-1} (x_d - x_{d-1}^2) + 2 (x_{d-1} - 1) \\ 200 (x_d - x_{d -1}^2)\end{pmatrix}.

Rosenbrock's banana function has one global minimum f_{\rm banana}(x^{\star}) = 0 at x^{\star} = (1,\dots, 1).

Value

banana returns the function value of Rosenbrock's banana function at x.

bananaGrad returns the gradient of Rosenbrock's banana 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")}.

Rosenbrock, H. H. (1960). An Automatic Method for Finding the Greatest or least Value of a Function. The Computer Journal, 3(3):175–184. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1093/comjnl/3.3.175")}.

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 Rosenbrock's banana function 
n.grid <- 50
x1 <- seq(-2, 2, length.out = n.grid)
x2 <- seq(-1, 3, length.out = n.grid)
y <- outer(x1, x2, function(x1, x2) banana(cbind(x1, x2)))
contour(x1, x2, y, xaxs = "i", yaxs = "i", nlevels = 25, xlab = "x1", ylab = "x2")

# Perspective plot of Rosenbrock's banana 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.