Hartman's Function No.01

I. Mathematical Expression:

$$f(X)=-\sum_{i=1}^{m}c_i \ e^{\displaystyle\left[-\sum_{j=1}^{n}a_{i,j}\left(x_j-p_{i,j}\right)^2\right]} \ \ , \ \ \ m=4, \ n=3$$

where:

\(\bullet\) The values of \(a\), \(c\) and \(p\) are tabulated in Table 1

\(\bullet\) \(0\leq x_j\leq 1\) , \(j=1,2,3\)

\(\bullet\) \(f_{min}(X^*)=-3.86278214782076\)

\(\bullet\) \(x^*_i\cong (0.1,0.55592003,0.85218259)\)

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] X. Yao, Y. Liu, and G. Lin, "Evolutionary Programming Made Faster," IEEE Transactions on Evolutionary Computation, vol. 3, no. 2, pp. 82-102, Jul. 1999.
[2] 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.
[3] A. Gavana, "Test Functions Index," Feb. 2013, [Accessed April 01, 2013]. [Online]. Available: http://infinity77.net/global_optimization/test_functions.html
[4] X. Zhao and X.-S. Gao, "Affinity Genetic Algorithm," Journal of Heuristics, vol. 13, no. 2, pp.133-150, Apr. 2007.
[5] 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
[6] R. Chelouah and P. Siarry, "Tabu Search Applied to Global Optimization," European Journal of Operational Research, vol. 123, no. 2, pp. 256-270, Jun. 2000.
[7] 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.
[8] J. B. Lee and B. C. Lee, "A Global Optimization Algorithm Based on the New Filled Function Method and the Genetic Algorithm," Engineering Optimization, vol. 27, no. 1, pp. 1-20, 1996.
[9] 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