Mathematical Definition

Description and Features

  • The function is continuous.
  • The function is not convex.
  • The function is defined on 3-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 $x_i \in [0, 2]$ for $i=1, ā€¦, 3$.

Global Minima

The global minima $f(\textbf{x}^{\ast})=0$ are located at $\mathbf{x^\ast}=(0, 0, 0)$.

Implementation

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

% Computes the value of the Wolfe function.
% SCORES = WOLFEFCN(X) computes the value of the Wolfe 
% function at point X. WOLFEFCN accepts a matrix of size M-by-3 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/wolfefcn
% 
% 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 = wolfefcn(x)
    n = size(x, 2);
    assert(n == 3, 'The Wolfe function is defined only on the 3-D space.')
    X = x(:, 1);
    Y = x(:, 2);
    Z = x(:, 3);
    
    scores = (4/3)*(((X .^ 2 + Y .^ 2) - (X .* Y)).^(0.75)) + Z;
end 

The function can be represented in Latex as follows:

f(x, y, z) = \frac{4}{3}(x^2 + y^2 - xy)^{0.75} + z

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. 150ā€“194 (2013), arXiv:1308.4008
  • H. P. Schwefel, ā€œNumerical Optimization for Computer Models,ā€ John Wiley Sons, 1981.