Benchmark test functions

The following benchmark functions are implemented in the submodule Newtman.TestFunctions. Each function is defined in the survey by Jamil and Yang^[1]. We explain them in detail here for quick reference purposes. No other information more than the solutions to each of the optimization problems is provided.

The purpose of these functions is to check the validity of the implementations in this package. By solving these benchmark optimization problems we can trust that the implemenations are correct and that they will give reasonable results in other similar problems. It is expected that virtually all implementations can solve these functions, or at least a considerable subset of these.

The previous point is very important. Due to the No Free Lunch theorem^[2] and its extension to metaheuristics^[3], no single optimization algorithm is better than another for a set of optimization problems. This is a very important result, and one of the main reasons why most of the time some algorithms tend to perform better than other for a given optimization problem.

`Sphere`

The Sphere function is defined as:

\[f(\mathbf{x}) = \sum_{i=1}^{d} x_i^2\]

with $d$ the dimension of the design vector $\mathbf{x}$, normally evaluated within the bounds $0 \leq x_i \leq 10$.

Solution

\[f(\mathbf{x^*}) = 0, \quad \mathbf{x^*} = (0, \cdots, 0)\]

`Easom`

The Easom function is defined as:

\[f(\mathbf{x}) = -\cos{(x_1)} \cos{(x_2)} \exp{[-(x_1 - \pi)^2 - (x_2 - \pi)^2]}\]

where the design vector is a 2-D vector only. It is normally evaluated within the range $-100 \leq x_i \leq 100$.

Solution

\[f(\mathbf{x^*}) = -1, \quad \mathbf{x^*} = (\pi, \pi)\]

`Ackley`

The Ackley function is defined as:

\[f(\mathbf{x}) = -20 \exp{\left[ -0.02 \sqrt{\frac{1}{d}\sum_{i=1}^{d}{x_i^2}} \right]} - \exp{\left[\frac{1}{d}\sum_{i=1}^{d}{\cos{(2 \pi x_i)}}\right]} + 20 + e\]

where the design vector is a d-dimensional vector. Normally evaluated within the range $-35 \leq x_i \leq 35$.

Solution

\[f(\mathbf{x^*}) = 0, \quad \mathbf{x^*} = (0, \cdots, 0)\]

`Rosenbrock`

The famous Rosenbrock function is defined as:

\[f(\mathbf{x}) = \sum_{i=1}^{N-1} \left[100(x_{i-1}-x_i^2)^2 +(1-x_i)^2 \right]\]

where the design vector is a N-dimensional vector. The search space range is normally $-\infty \leq x_i \leq \infty$.

Solution

\[f(\mathbf{x^*}) = 0, \quad \mathbf{x^*} = (1, \cdots, 1)\]

`GoldsteinPrice`

The Goldstein-Price function is defined as:

\[f(x,y)=[1 + (x + y + 1)^2(19 − 14x+3x^2− 14y + 6xy + 3y^2)] \times \\ [30 + (2x − 3y)^2(18 − 32x + 12x^2 + 4y − 36xy + 27y^2)]\]

where $x$ and $y$ are the elements of a $2D$ design vector.

Solution

\[f(\mathbf{x^*}) = 3, \quad \mathbf{x^*} = (0, -1)\]

`Beale`

The Beale function is defined as:

\[f(x, y) = (1.5-x+xy)^2+(2.25-x+xy^2)^2+(2.625-x+xy^3)^2\]

where $x$ and $y$ are the elements of a $2D$ design vector.

Solution

\[f(\mathbf{x^*}) = 0, \quad \mathbf{x^*} = (3, 0.5)\]

`Levy`

The Lévy function is defined as:

\[f(\mathbf{x}) = \sin^{2}{\pi w_1} + \sum_{i=1}^{d-1} (w_i-1)^2 [1+10\sin^{2}{\pi w_1 + 1}] + (w_d-1)^2 [1+\sin^{2}{2\pi w_d}]\]

where

\[w_i = 1 + \frac{x_i-1}{4}\]

and $d$ is the dimension of the vector.

Solution

\[f(\mathbf{x^*}) = 0, \quad \mathbf{x^*} = (1, \dots, 1)\]

References

1Jamil, M., & Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150–194. https://doi.org/10.1504/IJMMNO.2013.055204
2Wolpert, D. H. and Macready, W. G. (1997) ‘No free lunch theorems for optimization’, IEEE Transactions on Evolutionary Computation, 1(1), pp. 67–82. doi: 10.1109/4235.585893.
3Joyce, T. and Herrmann, J. M. (2018) ‘A Review of No Free Lunch Theorems, and Their Implications for Metaheuristic Optimisation’, in Yang, X.-S. (ed.) Nature-Inspired Algorithms and Applied Optimization. Cham: Springer International Publishing (Studies in Computational Intelligence), pp. 27–51. doi: 10.1007/978-3-319-67669-2_2.