Mathematical Definition

\[f(\mathbf{x})=f(x_1, ..., x_n)=\sum_{i=1}^{n}{ix_i^2}\]

Plots

Sum Squares Function

Sum Squares Function

Sum Squares Function

Sum Squares Function

A contour of the function is presented below:

Sum Squares Function

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 differentiable.
  • The function is separable.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x_i \in [-10, 10]$ 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 sumsquares

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

MATLAB

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

% Computes the value of the Sum Squares function.
% SCORES = SUMSQUARESFCN(X) computes the value of the Sum Squares
% function at point X. SUMSQUARESFCN 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 = sumsquaresfcn(x)
   
   [m, n] = size(x);
   x2 = x .^2;
   I = repmat(1:n, m, 1);
   scores = sum( I .* x2, 2);
   
end

The function can be represented in Latex as follows:

f(\mathbf{x})=f(x_1, ..., x_n)=\sum_{i=1}^{n}{ix_i^2}

References: