Mathematical Definition

Plots

Alpine N. 2 Function

Alpine N. 2 Function

Alpine N. 2 Function

Alpine N. 2 Function

A contour of the function is presented below:

Alpine N. 2 Function

Description and Features

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

Input Domain

The function can be defined on any positive input domain but it is usually evaluated on $x_i \in [0, 10]$ for $i=1, …, n$.

Global Minima

The function was devised By Clerc as a maximization problem and hence, the orginial paper gave $f(\textbf{x}^{\ast})=2.808^n$, located at $\mathbf{x^\ast}=(7.917, …, 7.917)$, as its global maximum. The function can be used for minization by negating its value.

Implementation

An implementation of the Alpine N. 2 Function with MATLAB is provided below.

% Computes the value of the Alpine N. 2 function.
% SCORES = ALPINEN2FCN(X) computes the value of the Alpine N. 2
% function at point X. ALPINEN2FCN 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:
% benchmarkfcns.xyz/fcns/alpinen2fcn
% See also: alpinen1fcn
% 
% 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 = alpinen2fcn(x)
     scores = prod(sqrt(x) .* sin(x), 2);
end 

The function can be represented in Latex as follows:

f(\mathbf x)=f(x_1, ..., x_n) = \prod_{i=1}^{n}\sqrt{x_i}sin(x_i)

See Also:

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
  • M. Clerc, “The Swarm and the Queen, Towards a Deterministic and Adaptive Particle Swarm Optimization, ” IEEE Congress on Evolutionary Computation, Washington DC, USA, pp. 1951-1957, 1999.