testFunctions function

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)

Arguments

• x: a numeric vector of arguments. See Details.

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

$$\exp(1) + 20 -20 \exp{\left(-0.2 \sqrt{\frac{1}{n}\sum_{i=1}^n x_i^2}\right)} - \exp{\left(\frac{1}{n}\sum_{i=1}^n \cos(2 \pi x_i)\right)}\,.exp(1) + 20 - 20 * exp(-0.2 * sqrt(sum(x^2)/n)) - exp(sum(cos(2 * pi * x))/n)$$

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

$$-(y + 47) \sin\left(\sqrt{|y + \frac{x}{2} + 47|}\right) -x \sin\left(\sqrt{|x - (y + 47)|}\right)\,.-(y + 47) * sin(sqrt(abs(y + x/2 + 47))) -x * sin(sqrt(abs(x - (y + 47))))$$

The minimum function value is -959.6407; reached at c(512, 404.2319).

The Griewank function is given by

$$1+\frac{1}{4000} \sum^n_{i=1} x_i^2 - \prod_{i=1}^n \cos \left(\frac{x_i}{\sqrt{i}}\right)\,.sum(x^2)/4000 - prod(cos(x/sqrt(1:n))) + 1$$

The function is minimised at $x = 0$; its minimum value is zero.

The Rastrigin function:

$$10n + \sum_{i=1}^n \left(x_i^2 -10\cos(2\pi x_i)\right)\,.10*n + sum(x^2 - 10 * cos(2 * pi * x))$$

The minimum function value is zero; reached at $x = 0$.

The Rosenbrock (or banana) function:

$$\sum_{i=1}^{n-1}\left(100 (x_{i+1}-x_i^2)^2 + (1-x_i)^2\right)\,.sum(100*(x[2:n] - x[1:(n - 1)]^2)^2 + (1 - x[1:(n - 1)])^2)$$

The minimum function value is zero; reached at $x = 1$.

The Schwefel function:

$$\sum_{i=1}^n \left(-x_i \sin\left(\sqrt{|x_i|}\right)\right)\,.sum(-x * sin(sqrt(abs(x))))$$

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

$$\exp(\sin(50x)) + \sin(60 e^y) + \sin(70 \sin(x)) + \sin(\sin(80y)) - \sin(10(x+y)) + \frac{1}{4}(x^2+y^2)\,.exp(sin(50 * x)) + sin(60 * exp(y)) + sin(70 * sin(x)) + sin(sin(80 * y)) - sin(10 * (x + y)) + (x^2 + y^2)/4$$

The minimum function value is -3.3069; reached at c(-0.0244, 0.2106).

Returns

The objective function evaluated at x (a numeric vector of length one).

References

Gilli, M., Maringer, D. and Schumann, E. (2019) Numerical Methods and Optimization in Finance. 2nd edition. Elsevier. tools:::Rd_expr_doi("10.1016/C2017-0-01621-X")

Schumann, E. (2023) Financial Optimisation with R (NMOF Manual). https://enricoschumann.net/NMOF.htm#NMOFmanual

Author(s)

Enrico Schumann

Warning

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.

See Also

DEopt, PSopt

Examples

## 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)