# Gear Function

I. Mathematical Expression:

$$f(X)=\left(\frac{\displaystyle 1.0}{\displaystyle 6.931}-\frac{\displaystyle \left \lfloor x_1 \right \rfloor \cdot \left \lfloor x_2 \right \rfloor}{\displaystyle \left \lfloor x_3 \right \rfloor \cdot \left \lfloor x_4 \right \rfloor}\right)^2$$

where:

$$\bullet$$ $$12\leq x_i\leq 60$$ , $$i=1,2,3,4$$

$$\bullet$$ Infinite number of global minimum: $$f_{min}(X^*)=2.700857148886513 \times 10^{-12}$$

$$\bullet$$ $$\left \lfloor x^*_i \right \rfloor=(16, 19, 43, 49)$$ $$\rightarrow$$ $$\{x_1 , x_2\}$$ and $$\{x_3 , x_4\}$$ can be permuted

II. References:

[1] E. P. Adorio, "MVF - Multivariate Test Functions Library in C for Unconstrained Global Optimization," Quezon City, Metro Manila, Philippines, Jan. 2005. [Online]. Available: http://geocities.ws/eadorio/mvf.pdf
[2] A. Gavana, "Test Functions Index," Feb. 2013, [Accessed April 01, 2013]. [Online]. Available: http://infinity77.net/global_optimization/test_functions.html
[3] 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