Mathematical Definition

Plots

Alpine N. 1 Function

Alpine N. 1 Function

Alpine N. 1 Function

Alpine N. 1 Function

Alpine N. 1 Function

Alpine N. 1 Function

Alpine N. 1 Function

Alpine N. 1 Function

Two contours of the function are presented below:

Alpine N. 1 Function

Alpine N. 1 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 has a global minimum $f(\textbf{x}^{\ast})=0$ located at $\mathbf{x^\ast}=(0, …, 0)$.

Implementation

An implementation of the Alpine N. 1 Function with MATLAB is provided below.

% Computes the value of the Alpine N. 1 function.
% SCORES = ALPINEN1FCN(X) computes the value of the Alpine N. 1
% function at point X. ALPINEN1FCN 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/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 = alpinen1fcn(x)
     scores = sum(abs(x .* sin(x) + 0.1 * x), 2);
end 

The function can be represented in Latex as follows:

f(\mathbf x)=f(x_1, ..., x_n) = \sum{i=1}^{n}|x_i sin(x_i)+0.1x_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.