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. 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] 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