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I. Mathematical Expression:

$$f(X)=\gamma^2(X)+\sum_{k=1}^{4}\left(\alpha^2_k(X)+\beta^2_k(X)\right)$$

where:

\(\bullet\) \(\alpha_k(X)=\left(1-x_1x_2\right)x_3\left[e^{\displaystyle x_5\left(g_{1,k}-g_{3,k} \ x_7 \times 10^{-3} -g_{5,k} \ x_8 \times 10^{-3}\right)}-1\right]-g_{5,k}+g_{4,k} \ x_2\)

\(\bullet\) \(\beta_k(X)=\left(1-x_1x_2\right)x_4\left[e^{\displaystyle x_6\left(g_{1,k}-g_{2,k}-g_{3,k} \ x_7 \times 10^{-3} +g_{4,k} \ x_9 \times 10^{-3}\right)}-1\right]-g_{5,k} \ x_1+g_{4,k}\)

\(\bullet\) \(\gamma(X)=x_1x_3-x_2x_4\)

\(\bullet\) The values of \(g\) are tabulated in Table 1

\(\bullet\) \(-10\leq x_i\leq 10\) , \(i=1,2,\cdots,9\)

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

\(\bullet\) \(x^*_i=(0.9,0.45,1,2,8,8,5,1,2)\)

 

II. References:

[1] M. M. Ali, C. Khompatraporn, and Z. B. Zabinsky, "A Numerical Evaluation of Several Stochastic Algorithms on Selected Continuous Global Optimization Test Problems," Journal of Global Optimization, vol. 31, no. 4, pp. 635-672, Apr. 2005.
[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

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