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I. Mathematical Expression:

$$f(X)=\left[100\left(x_2-x^2_1\right)\right]^2+\left(1-x_1\right)^2+90\left(x_4-x^2_3\right)^2+\left(1-x_3\right)^2+$$

$$10.1\left[\left(x_2-1\right)^2+\left(x_4-1\right)^2\right]+19.8\left(x_2-1\right)\left(x_4-1\right)$$

where:

\(\bullet\) \(-10\leq x_i\leq 10\) , \(i=1,2,3,4\)

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

\(\bullet\) \(x^*_i=1\)

 

II. References:

[1] S. S. Rao, Engineering Optimization: Theory and Practice, 4th ed. Hoboken, New Jersey: John Wiley & Sons, 2009.
[2] X. Zhao and X.-S. Gao, "Affinity Genetic Algorithm," Journal of Heuristics, vol. 13, no. 2, pp.133-150, Apr. 2007.
[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] S. Rahnamayan, H. R. Tizhoosh, and M. M. A. Salama, "A Novel Population Initialization Method for Accelerating Evolutionary Algorithms," Computers & Mathematics with Applications, vol. 53, no. 10, pp. 1605-1614, May 2007.
[5] A. D. Belegundu and T. R. Chandrupatla, Optimization Concepts and Applications in Engineering, 2nd ed. New York: Cambridge University Press, 2011.
[6] M. Jamil and X. S. Yang, "A Literature Survey of Benchmark Functions for Global Optimization Problems," International Journal of Mathematical Modelling and Numerical Optimisation, vol. 4, no. 2, pp. 150–194, Aug. 2013.
[7] A. Gavana, "Test Functions Index," Feb. 2013, [Accessed April 01, 2013]. [Online]. Available: http://infinity77.net/global_optimization/test_functions.html
[8] 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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