# Powell Sum Function

# Mathematical Definition

# 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

An implementation of the **Powell Sum Function** with `MATLAB`

is provided below.

The function can be represented in Latex as follows:

# 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.