# 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. 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