Mathematical Definition

\[f(\mathbf x)=f(x_1, ..., x_n)=1-cos(2\pi\sqrt{\sum_{i=1}^{D}x_i^2})+0.1\sqrt{\sum_{i=1}^{D}x_i^2}\]

Plots

Salomon Function

Salomon Function

Salomon Function

Salomon Function

Contour of the function is presented below:

Salomon Function

Description and Features

  • The function is continuous.
  • The function is not convex.
  • The function is defined on n-dimensional space.
  • The function is multimodal.
  • The function is differentiable.
  • The function is non-separable.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x_i \in [-100, 100]$ for $i=1, …, n$.

Global Minima

The function has one global minimum $f(\textbf{x}^{\ast})=0$ at $\textbf{x}^{\ast} = (0, …, 0)$.

Implementation

Python

For Python, the function is implemented in the benchmarkfcns package, which can be installed from command line with pip install benchmarkfcns.

from benchmarkfcns import salomon

print(salomon([[0, 0, 0],
              [1, 1, 1]]))

MATLAB

An implementation of the Salomon Function with MATLAB is provided below.

% Computes the value of the Salomon's benchmark function.
% SCORES = SALOMONFCN(X) computes the value of the Salomon's   
% function at point X. SALOMONFCN accepts a matrix of size M-by-N  
% and returns a vetor SCORES of size M-by-1 in which each row contains the 
% function value for the corresponding row of X.
% For more information please visit: 
% https://en.wikipedia.org/wiki/Test_functions_for_optimization
% 
% Author: Mazhar Ansari Ardeh
% Please forward any comments or bug reports to mazhar.ansari.ardeh at
% Google's e-mail service or feel free to kindly modify the repository.
function scores = salomonfcn(x)
    x2 = x .^ 2;
    sumx2 = sum(x2, 2);
    sqrtsx2 = sqrt(sumx2);
    
    scores = 1 - cos(2 .* pi .* sqrtsx2) + (0.1 * sqrtsx2);
end

The function can be represented in Latex as follows:

f(\mathbf x)=f(x_1, ..., x_n)=1-cos(2\pi\sqrt{\sum_{i=1}^{D}x_i^2})+0.1\sqrt{\sum_{i=1}^{D}x_i^2}

References:

  • Momin Jamil and Xin-She Yang, A literature survey of benchmark functions for global optimization problems, Int. Journal of Mathematical Modelling and Numerical Optimisation}, Vol. 4, No. 2, pp. 150–194 (2013), arXiv:1308.4008
  • http://profesores.elo.utfsm.cl/~tarredondo/info/soft-comp/functions/node12.html
  • R. Salomon, “Re-evaluating Genetic Algorithm Performance Under Corodinate Rotation of Benchmark Functions: A Survey of Some Theoretical and Practical Aspects of Genetic Algorithms,” BioSystems, vol. 39, no. 3, pp. 263-278, 1996.