Mathematical Definition

Plots

Three-Hump Camel Function

Three-Hump Camel Function

Three-Hump Camel Function

Three-Hump Camel Function

Three-Hump Camel Function

Two contours of the function are as presented below:

Three-Hump Camel Function

Three-Hump Camel Function

Description and Features

  • The function is continuous.
  • The function is not convex.
  • The function is defined on 2-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 $x_i \in [-5, 5]$ for $i=1, 2$.

Global Minima

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

Implementation

An implementation of the Three-Hump Camel Function with MATLAB is provided below.

% Computes the value of the Three-hump camel benchmark function.
% SCORES = THREEHUMPCAMELFCN(X) computes the value of the Three-hump camel   
% function at point X. THREEHUMPCAMELFCN accepts a matrix of size M-by-2  
% 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 = threehumpcamelfcn(x)
    
    n = size(x, 2);
    assert(n == 2, 'The Three-hump camel function is only defined on a 2D space.')
    X = x(:, 1);
    Y = x(:, 2);
    
    scores = (2 * X .^ 2) - (1.05 * (X .^ 4)) + ((X .^ 6) / 6) + X .* Y + Y .^2;
end

The function can be represented in Latex as follows:

f(x,y)=2x^2-1.05x^4+\frac{x^6}{6}+xy+y^2

References: