Mathematical Definition

\[f(x, y)=sin(x + y) + (x - y) ^2 - 1.5x + 2.5 y + 1\]

Plots

McCormick Function

McCormick Function

McCormick Function

The contour of the function is as presented below:

McCormick Function

Description and Features

  • The function is continuous.
  • The function is convex.
  • The function is defined on 2-dimensional space.
  • The function is multimodal.
  • The function is differentiable.
  • The function is .

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x \in [-1.5, 4]$ and $y \in [-3, 3]$ .

Global Minima

The function has one global minimum $f(\textbf{x}^{\ast})\approx −1.9133$ at $\textbf{x}^{\ast} = (−0.547,−1.547)$.

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 mccormick

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

MATLAB

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

% Computes the value of the McCormick benchmark function.
% SCORES = MCCORMICKFCN(X) computes the value of the McCormick function 
% at point X. MCCORMICKFCN 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 = mccormickfcn(x)
    
    n = size(x, 2);
    assert(n == 2, 'The McCormick function is only defined on a 2D space.')
    X = x(:, 1);
    Y = x(:, 2);
    
    scores = sin(X + Y) + ((X - Y) .^2 ) - 1.5 * X + 2.5 * Y + 1;
end

The function can be represented in Latex as follows:

f(x, y)=sin(x + y) + (x - y) ^2 - 1.5x + 2.5 y + 1

References: