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I. Mathematical Expression:

$$f(X)=\sum_{i=1}^{n}\begin{cases}
\displaystyle 0.15d_i \left[z_i-0.05 \ \text{sign}\left(z_i\right)\right]^2 & \text{ if } \left|x_i-z_i\right|<0.05 \\
d_ix^2_i & \text{ otherwise }
\end{cases}$$

where:

\(\bullet\) \(z_i=0.2\left \lfloor \left|\frac{x_i}{s_i}\right|+0.49999 \right \rfloor \text{sign}\left(x_i\right)\)

\(\bullet\) \(s_i=0.2 \ , d_i=[1, \ 1000, \ 10, \ 100]\)

\(\bullet\) \(-100\leq x_i\leq 100\) , \(i=1,2,3,4\)

\(\bullet\) \(f_{min}(X^*)=0\)

\(\bullet\) \(x^*_i=0\)

II. References:

[1] R. Storn and K. Price, "Differential Evolution - A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces," Journal of Global Optimization, vol. 11, no. 4, pp. 341-359, Dec. 1997.
[2] A. Gavana, "Test Functions Index," Feb. 2013, [Accessed April 01, 2013]. [Online]. Available: http://infinity77.net/global_optimization/test_functions.html
[3] 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
[4] 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

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