# 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. 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