Classical Test Functions for Unconstrained Optimisation
A number of functions that have been suggested in the literature as benchmarks for unconstrained optimisation.
tfAckley(x) tfEggholder(x) tfGriewank(x) tfRastrigin(x) tfRosenbrock(x) tfSchwefel(x) tfTrefethen(x)
x |
a numeric vector of arguments. See Details. |
All functions take as argument only one variable, a
numeric vector x whose length determines the
dimensionality of the problem.
The Ackley function is implemented as
exp(1) + 20 - 20 * exp(-0.2 * sqrt(sum(x^2)/n)) - exp(sum(cos(2 * pi * x))/n)
The minimum function value is zero; reached at x = 0.
The Eggholder takes a two-dimensional x,
here written as x and y. It is defined as
-(y + 47) * sin(sqrt(abs(y + x/2 + 47))) - x * sin(sqrt(abs(x - (y + 47))))
The minimum function value is -959.6407; reached at c(512, 404.2319).
The Griewank function is given by
sum(x^2)/4000 - prod(cos(x/sqrt(1:n))) + 1
The function is minimised at x = 0; its minimum value is zero.
The Rastrigin function:
10*n + sum(x^2 - 10 * cos(2 * pi * x))
The minimum function value is zero; reached at x = 0.
The Rosenbrock (or banana) function:
sum(100*(x[2:n] - x[1:(n - 1)]^2)^2 + (1 - x[1:(n - 1)])^2)
The minimum function value is zero; reached at x = 1.
The Schwefel function:
sum(-x * sin(sqrt(abs(x))))
The minimum function value (to about 8 digits) is -418.9829n; reached at x = 420.9687.
Trefethen's function takes a two-dimensional x (here written as x and y); it is defined as
exp(sin(50 * x)) + sin(60 * exp(y)) + sin(70 * sin(x)) + sin(sin(80 * y)) - sin(10 * (x + y)) + (x^2 + y^2)/4
The minimum function value is -3.3069; reached at c(-0.0244, 0.2106).
The objective function evaluated at x (a numeric vector of
length one).
These test functions represent artificial problems. It is practically not too helpful to fine-tune a method on such functions. (That would be like memorising all the answers to a particular multiple-choice test.) The functions' main purpose is checking the numerical implementation of algorithms.
Enrico Schumann
Gilli, M., Maringer, D. and Schumann, E. (2019) Numerical Methods and Optimization in Finance. 2nd edition. Elsevier. https://www.elsevier.com/books/numerical-methods-and-optimization-in-finance/gilli/978-0-12-815065-8
Schumann, E. (2019) Financial Optimisation with R (NMOF Manual). http://enricoschumann.net/NMOF.htm#NMOFmanual
## persp for two-dimensional x
## Ackley
n <- 100L; surf <- matrix(NA, n, n)
x1 <- seq(from = -2, to = 2, length.out = n)
for (i in 1:n)
for (j in 1:n)
surf[i, j] <- tfAckley(c(x1[i], x1[j]))
persp(x1, x1, -surf, phi = 30, theta = 30, expand = 0.5,
col = "goldenrod1", shade = 0.2, ticktype = "detailed",
xlab = "x1", ylab = "x2", zlab = "-f", main = "Ackley (-f)",
border = NA)
## Trefethen
n <- 100L; surf <- matrix(NA, n, n)
x1 <- seq(from = -10, to = 10, length.out = n)
for (i in 1:n)
for (j in 1:n)
surf[i, j] <- tfTrefethen(c(x1[i], x1[j]))
persp(x1, x1, -surf, phi = 30, theta = 30, expand = 0.5,
col = "goldenrod1", shade = 0.2, ticktype = "detailed",
xlab = "x1", ylab = "x2", zlab = "-f", main = "Trefethen (-f)",
border = NA)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.