Mathematical Definition

\[f(x,y)=[1 + (x + y + 1)^2(19 − 14x+3x^2− 14y + 6xy + 3y^2)][30 + (2x − 3y)^2(18 − 32x + 12x^2 + 4y − 36xy + 27y^2)]\]

Plots

Goldstein-Price Function

The contour of the function is as presented below:

Goldstein-Price 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 .
  • The function is non-scalable.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x \in [-2, 2]$ and $y \in [-2, 2]$ .

Global Minima

The function has four global minima $f(\textbf{x}^{\ast})=3$ at $\textbf{x}^{\ast} = (0, -1)$.

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 goldsteinprice

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

MATLAB

An implementation of the Goldstein-Price Function with MATLAB is provided below.

% Computes the value of GOldstein-Price benchmark function.
% SCORES = GOLDSTEINPRICEFCN(X) computes the value of the GOLDSTEINPRICEFCN  
% function at point X. GOLDSTEINPRICEFCN 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 = goldsteinpricefcn(x)
    n = size(x, 2);
    assert(n == 2, 'The Goldstein-Price function is only defined on a 2D space.')
    X = x(:, 1);
    Y = x(:, 2);
    
    scores = (1 + ((X + Y + 1).^2) * (19 - (14 * X) + (3 * (X .^2)) - 14*Y + (6 .* X.*Y) + (3 * (Y.^2)))) .* ...
        (30 + ((2 * X - 3 * Y).^2) .* (18 - 32 * X + 12 * (X .^2) + 48 * Y - (36 .* X.*Y) + (27 * (Y.^2))) );
end

The function can be represented in Latex as follows:

f(x,y)=[1 + (x + y + 1)^2(19 − 14x+3x^2− 14y + 6xy + 3y^2)][30 + (2x − 3y)^2(18 − 32x + 12x^2 + 4y − 36xy + 27y^2)]

References: