ANNs XOR Function

I. Mathematical Expression:

$$f(X)=f_1(X)+f_2(X)+f_3(X)+f_4(X)$$

where:

\(\bullet\) \(f_1(X)=\left\{1+e^{\displaystyle\left[-\frac{x_7}{1+e^{\displaystyle-\left(x_1+x_2+x_5\right)}}-\frac{x_8}{1+e^{\displaystyle-\left(x_3+x_4+x_6\right)}}-x_9\right]}\right\}^{-2}\)

\(\bullet\) \(f_2(X)=\left\{1+e^{\displaystyle\left[-\frac{x_7}{1+e^{\displaystyle-x_5}}-\frac{x_8}{1+e^{\displaystyle-x_6}}-x_9\right]}\right\}^{-2}\)

\(\bullet\) \(f_3(X)=\left\{1-\frac{1}{1+e^{\displaystyle\left[-\frac{x_7}{1+e^{\displaystyle-\left(x_1+x_5\right)}}-\frac{x_8}{1+e^{\displaystyle-\left(x_3+x_6\right)}}-x_9\right]}}\right\}^2\)

\(\bullet\) \(f_4(X)=\left\{1-\frac{1}{1+e^{\displaystyle\left[-\frac{x_7}{1+e^{\displaystyle-\left(x_2+x_5\right)}}-\frac{x_8}{1+e^{\displaystyle-\left(x_4+x_6\right)}}-x_9\right]}}\right\}^2\)

\(\bullet\) \(-1\leq x_i\leq 1\) , \(i=1,2,\cdots,9\)

\(\bullet\) \(f_{min}(X^*)\approx 0.959759\)

\(\bullet\) \(x^*_i\approx (0.99999,0.99993,-0.89414,0.99994,0.55932,0.99994,0.99994,-0.99963,-0.08272)\)

II. Citation Policy:

If you publish material based on databases obtained from this repository, then, in your acknowledgments, please note the assistance you received by using this repository. This will help others to obtain the same data sets and replicate your experiments. We suggest the following pseudo-APA reference format for referring to this repository:

Ali R. Al-Roomi (2015). Unconstrained Single-Objective Benchmark Functions Repository [https://www.al-roomi.org/benchmarks/unconstrained]. Halifax, Nova Scotia, Canada: Dalhousie University, Electrical and Computer Engineering.

Here is a BiBTeX citation as well:

@MISC{Al-Roomi2015,
author = {Ali R. Al-Roomi},
title = {{Unconstrained Single-Objective Benchmark Functions Repository}},
year = {2015},
address = {Halifax, Nova Scotia, Canada},
institution = {Dalhousie University, Electrical and Computer Engineering},
url = {https://www.al-roomi.org/benchmarks/unconstrained}
}

III. References:

[1] S. Mishra, "Repulsive Particle Swarm Method on Some Difficult Test Problems of Global Optimization," Shillong, India, Oct. 2006. [Online]. Available: http://mpra.ub.uni-muenchen.de/1742/1/MPRA_paper_1742.pdf
[2] Ali R. Alroomi, "The Farm of Unconstrained Benchmark Functions," University of Bahrain, Electrical and Electronics Department, Bahrain, Oct. 2013. [Online]. Available: http://www.al-roomi.org/cv/publications