Mathematical Definition

\[f(x, y) = (x^2 + y - 10)^2 + (x + y^2 - 7)^2 + (x^2 + y^3 - 1)^2\]

Plots

Description and Features

• The function is continuous.
• The function is not convex.
• The function is defined on 2-dimensional space.
• The function is multimodal.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x_i \in [-500, 500]$ for all $i = 1, 2$.

Global Minima

The function has one global minimum at: $f({x}^{\ast}, {y}^{\ast})=0$ at $({x}^{\ast}, {y}^{\ast}) = (3.4091868222,-2.1714330361)$.

Implementation

Python

For Python, the function is implemented in the benchmarkfcns package and can be installed from command line with pip install benchmarkfcns.

from benchmarkfcns import elattar

print(elattar([[3.4091868222,-2.1714330361]]))

MATLAB

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 elattar

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

MATLAB

An implementation of the El-Attar-Vidyasagar-Dutta Function with MATLAB is provided below.

% Computes the value of the El-Attar function.
% SCORES = ELATTARFCN(X) computes the value of the El-Attar
% function at point X. ELATTARFCN 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 = elattarfcn(x)
    n = size(x, 2);
    assert(n == 2, 'The El-Attar et al. function is defined only on the 2-D space.')

    X = x(:, 1);
    Y = x(:, 2);

    scores = (X .^ 2 + Y - 10) .^ 2 + (X + Y .^ 2 - 7) .^ 2 + (X .^ 2 + Y .^ 3 - 1) .^ 2;
end

The function can be represented in Latex as follows:

$$f(x, y) = (x^2 + y - 10)^2 + (x + y^2 - 7)^2 + (x^2 + y^3 - 1)^2$$

References:

• R. A. El-Attar, M. Vidyasagar, S. R. K. Dutta, An Algorithm for II-norm Minimization With Application to Nonlinear II-approximation, SIAM Journal on Numverical Analysis, vol. 16, no. 1, pp. 70-86, 1979.