Mathematical Definition

Plots

Exponential Function

Exponential Function

Exponential Function

Exponential Function

A contour of the function is presented below:

Exponential 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 non-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 $f(\textbf{x}^{\ast})=$ at $\mathbf{x^\ast}=0$.

Implementation

An implementation of the Exponential Function with MATLAB is provided below.

% Computes the value of the Exponential function.
% SCORES = EXPONENTIALFCN(X) computes the value of the Exponential
% function at point X. EXPONENTIALFCN 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 = exponentialfcn(x)
   x2 = x .^2;
   
   scores = -exp(-0.5 * sum(x2, 2));
end

The function can be represented in Latex as follows:

f(\mathbf{x})=f(x_1, ..., x_n)=-exp(-0.5\sum_{i=1}^n{x_i^2})

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, “Opposition-Based Differential Evolution (ODE) with Variable Jumping Rate,” IEEE Sympousim Foundations Computation Intelligence, Honolulu, HI, pp. 81-88, 2007.