Mathematical Definition

\[f(\mathbf{x})=f(x_1, ..., x_n)=\sum_{i=1}^{n}|x_i|^{i+1}\]

Plots

A contour of the function is presented below:

Description and Features

• The function is continuous.
• The function is convex.
• The function is defined on n-dimensional space.
• The function is unimodal.
• The function is non-differentiable.
• The function is separable.

Input Domain

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

Global Minima

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

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 powellsum

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

MATLAB

An implementation of the Powell Sum Function with MATLAB is provided below.

% Computes the value of the Powell Sum benchmark function.
% SCORES = POWELLSUMFCN(X) computes the value of the Powell Sum function at 
% point X. POWELLSUMFCN accepts a matrix of size M-by-N 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 = powellsumfcn(x)
    n = size(x, 2);
    absx = abs(x);
    
    scores = 0;
    for i = 1:n
        scores = scores + (absx(:, i) .^ (i + 1));
    end
end

The function can be represented in Latex as follows:

f(\mathbf{x})=f(x_1, ..., x_n)=\sum_{i=1}^{n}|x_i|^{i+1}

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
• S. Rahnamyan, H. R. Tizhoosh, N. M. M. Salama, “A Novel Population Initialization Method for Accelerating Evolutionary Algorithms,” Computers and Mathematics with Applications, vol. 53, no. 10, pp. 1605-1614, 2007.
• Mukhopadhyay, Sumitra; Das, Soumyadip, (2016), A System on Chip Development of Customizable GA Architecture for Real Parameter Optimization Problem, in Handbook of Research on Natural Computing for Optimization Problems, IGI Global.