Pintér's Function No.01

I. Mathematical Expression:

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

where:

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

\(\bullet\) \(f_2(X)=\sum^{n-1}_{i=2} \left(x_{i-1}+5\sin\left(x_i\right)+x^2_{i+1}\right)^2\)

\(\bullet\) \(f_3(X)=\sum^{n-1}_{i=2} \ln^2 \left(1+\left|i \sin^2\left(x_{i-1}\right)+2x_i+3x_{i+1}\right|\right)\)

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

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

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

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

\(\bullet\) This benchmark function is presented in [1] with a specific search space up to \(n=50\). 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 [2] as \(\sum^{n-1}_{i=2}\). Based on the previous reason, there is no plot for this benchmark function.

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] 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