Mathematical Definition

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

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 $i=1, …, 3$.

Global Minima

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

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 wolfe

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

MATLAB

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.
% 
% 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. 150–194 (2013), arXiv:1308.4008
• H. P. Schwefel, “Numerical Optimization for Computer Models,” John Wiley Sons, 1981.