Cola Function

I. Mathematical Expression:

$$f(X,Y)=\sum_{j<i}\left(r_{i,j}-d_{i,j}\right)^2$$

where:

\(\bullet\) \(x_1=y_1=y_2=0\) , \(x_2=u_1\) , \(x_i=u_{2(i-2)}\) , \(y_i=u_{2(i-2)+1}\)

\(\bullet\) \(r_{i,j}\) is given by:

$$r_{i,j}=\sqrt{\left(x_i-x_j\right)^2+\left(y_i-y_j\right)^2}$$

\(\bullet\) \(d\) is a symmetric matrix given by:

$$d=[d_{i,j}]=\begin{Bmatrix}
\cdots & & & & & & & & & \\
1.27 & \cdots & & & & & & & & \\
1.69 & 1.43 & \cdots & & & & & & & \\
2.04 & 2.35 & 2.43 & \cdots & & & & & & \\
3.09 & 3.18 & 3.26 & 2.85 & \cdots & & & & & \\
3.20 & 3.22 & 3.27 & 2.88 & 1.55 & \cdots & & & & \\
2.86 & 2.56 & 2.58 & 2.59 & 3.12 & 3.06 & \cdots & & & \\
3.17 & 3.18 & 3.18 & 3.12 & 1.31 & 1.64 & 3.00 & \cdots & & \\
3.21 & 3.18 & 3.18 & 3.17 & 1.70 & 1.36 & 2.95 & 1.32 & \cdots & \\
2.38 & 2.31 & 2.42 & 1.94 & 2.85 & 2.81 & 2.56 & 2.91 & 2.97 & \cdots
\end{Bmatrix}$$

\(\bullet\) \(0\leq u_1\leq 4\) , \(-4\leq u_i\leq 4\) , \(i=2,3,\cdots,n\) , \(n=17\)

\(\bullet\) \(f_{min}(X^*)\approx 11.7464\)

\(\bullet\) \(u^*_i\approx (0.651906,1.30194,0.099242,-0.883791,-0.8796,\)
               \(0.204651,-3.28414, 0.851188,-3.46245, 2.53245,-0.895246,\)
               \(1.40992,-3.07367, 1.96257,-2.97872,-0.807849,-1.68978)\)

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] Kaj Madsen, and Julius Žilinskas, "Testing Branch-and-Bound Methods for Global Optimization," Kaunas University of Technology, Department of Informatics, Kaunas, Lithuania, May. 2000. [Online]. Available: http://www.imm.dtu.dk/documents/ftp/tr00/tr05_00.pdf
[2] 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
[3] 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