# Powell's Quartic Function (or Powell’s Singular Function)

I. Mathematical Expression:

$$f(X)=\left(x_1+10x_2\right)^2+5\left(x_3-x_4\right)^2+\left(x_2-2x_3\right)^4+10\left(x_1-x_4\right)^4$$

where:

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

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

$$\bullet$$ $$x^*_i=0$$

II. References:

[1] D. Whitley, S. Rana, J. Dzubera, and K. Mathias, "Evaluating Evolutionary Algorithms," Artificial Intelligence, vol. 85, no. 1-2, pp. 245-276, Aug. 1996.
[2] S. S. Rao, Engineering Optimization: Theory and Practice, 4th ed. Hoboken, New Jersey: John Wiley & Sons, 2009.
[3] 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.
[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