Pintér's Function No.02

I. Mathematical Expression:

$$f(X)=f_1(X)+f_2(X)+f_3(X)$$

where:

\(\bullet\) \(f_1(X)=\sum^n_{i=1} i x_i^2\)

\(\bullet\) \(f_2(X)=i\sin^2\left(x_{i-1}\sin\left(x_i\right)-x_i+\sin\left(x_{i+1}\right)\right)\)

\(\bullet\) \(f_3(X)=i\ln\left[1+i\left(x^2_{i-1}-2x_i+3x_{i+1}-\cos\left(x_i\right)+1\right)^2\right]\)

\(\bullet\) \(a_i\leq x_i\leq b_i\) , \(i=1,2,\cdots,n\) :

\(\begin{matrix}
a_1=-8.8 & a_{11}=-7.8 & a_{21}=-8.5 & a_{31}=-3.5 & a_{41}=-5.5 \\
b_1= \ \ 1.4 & b_{11}= \ \ 3.2 & b_{21}= \ \ 7.2 & b_{31}= \ \ 1.9 & b_{41}= \ \ 2.2 \\
a_2=-6.2 & a_{12}=-5.2 & a_{22}=-1.2 & a_{32}=-6.2 & a_{42}=-3.2 \\
b_2= \ \ 0.9 & b_{12}= \ \ 3.9 & b_{22}= \ \ 4.9 & b_{32}= \ \ 3.9 & b_{42}= \ \ 4.9 \\
a_3=-8.7 & a_{13}=-6.1 & a_{23}=-5.7 & a_{33}=-7.1 & a_{43}=-4.3 \\
b_3= \ \ 1.7 & b_{13}= \ \ 1.8 & b_{23}= \ \ 3.5 & b_{33}= \ \ 3.5 & b_{43}= \ \ 7.8 \\
a_4=-7.7 & a_{14}=-2.7 & a_{24}=-7.7 & a_{34}=-7.7 & a_{44}=-4.7 \\
b_4= \ \ 0.8 & b_{14}= \ \ 4.2 & b_{24}= \ \ 1.5 & b_{34}= \ \ 4.8 & b_{44}= \ \ 2.5 \\
a_5=-3.2 & a_{15}=-5.6 & a_{25}=-8.6 & a_{35}=-5.6 & a_{45}=-3.6 \\
b_5= \ \ 5.3 & b_{15}= \ \ 3.3 & b_{25}= \ \ 5.3 & b_{35}= \ \ 2.3 & b_{45}= \ \ 8.3 \\
a_6=-3.5 & a_{16}=-7.1 & a_{26}=-9.5 & a_{36}=-6.5 & a_{46}=-4.5 \\
b_6= \ \ 7.9 & b_{16}= \ \ 2.9 & b_{26}= \ \ 6.8 & b_{36}= \ \ 2.8 & b_{46}= \ \ 1.9 \\
a_7=-5.1 & a_{17}=-2.1 & a_{27}=-5.1 & a_{37}=-5.1 & a_{47}=-4.1 \\
b_7= \ \ 8.7 & b_{17}= \ \ 6.7 & b_{27}= \ \ 3.7 & b_{37}= \ \ 8.7 & b_{47}= \ \ 1.7 \\
a_8=-2.2 & a_{18}=-5.2 & a_{28}=-6.7 & a_{38}=-3.2 & a_{48}=-7.2 \\
b_8= \ \ 4.7 & b_{18}= \ \ 3.7 & b_{28}= \ \ 1.7 & b_{38}= \ \ 1.7 & b_{48}= \ \ 3.2 \\
a_9=-9.1 & a_{19}=-4.1 & a_{29}=-4.1 & a_{39}=-5.1 & a_{49}=-4.1 \\
b_9= \ \ 3.8 & b_{19}= \ \ 2.8 & b_{29}= \ \ 1.8 & b_{39}= \ \ 1.8 & b_{49}= \ \ 1.8 \\
a_{10}=-6.3 & a_{20}=-7.3 & a_{30}=-4.3 & a_{40}=-3.3 & a_{50}=-5.3 \\
b_{10}= \ \ 1.7 & b_{20}= \ \ 4.7 & b_{30}= \ \ 6.7 & b_{40}= \ \ 7.7 & b_{50}= \ \ 1.3
\end{matrix}\)

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

\(\bullet\) \(x^*_i =0\)

\(\bullet\) This benchmark function is presented in [1] with a specific search space up to \(n=50\). However, \(x_i \in [-10,10]\) is considered in [2, 3], so it can be used for any dimension. Because there are two terms (\(x_{i-1}\) and \(x_{i+1}\)) in \(f_2\) and \(f_3\), so the lowest accepted dimension is \(n=3\), and thus the summation limits are corrected in [4] as \(\sum^{n-1}_{i=2}\). Based on the previous reason, there is no plot for this benchmark function. This means that the 3D-plot in [3] is not correct.

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] János D. Pintér, Global Optimization in Action - Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications. Dordrecht, The Netherlands: Kluwer Academic Publishers, 1996.
[2] 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.
[3] A. Gavana, "Test Functions Index," Feb. 2013, [Accessed April 01, 2013]. [Online]. Available: http://infinity77.net/global_optimization/test_functions.html
[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