Mathematical Definition

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

Plots

Levi N. 13 Function

The contour of the function is as presented below:

Levi N. 13 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 non-separable.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x \in [-10, 10]$ and $y \in [-10, 10]$ .

Global Minima

The function has one global minimum $f(\textbf{x}^{\ast})=0$ at $\textbf{x}^{\ast} = (1, 1)$.

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 levin13

print(levin13([[0, 0],
              [1, 1]]))

MATLAB

An implementation of the Levi N. 13 Function with MATLAB is provided below.

% Computes the value of the Levi N. 13 benchmark function.
% SCORES = LEVIN13FCN(X) computes the value of the Levi N. 13 function at 
% point X. LEVIN13FCN 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 = levin13fcn(x)
    n = size(x, 2);
    assert(n == 2, 'Levi''s function is only defined on a 2D space.')
    X = x(:, 1);
    Y = x(:, 2);
    scores = sin(3 * pi * X) .^ 2 + ...
        ((X - 1).^2) .* (1 + sin(3 * pi * Y) .^ 2) + ...
        ((Y - 1).^2) .* (1 + sin(2 * pi * Y) .^ 2);
end

The function can be represented in Latex as follows:

f(x, y) = sin^2(3\pi x)+(x-1)^2(1+sin^2(3\pi y))+(y-1)^2(1+sin^2(2\pi y))