I. Mathematical Expression:

$$\text{VNT}:\begin{cases} \text{Minimize } & f_1(X)=0.5\left(x_1^2+x_2^2\right)+\sin\left(x_1^2+x_2^2\right)\\ \text{Minimize } & f_2(X)=\frac{\displaystyle \left(3x_1-2x_2+4\right)^2}{\displaystyle 8}+\frac{\displaystyle \left(x_1-x_2+1\right)^2}{\displaystyle 27}+15\\ \text{Minimize } & f_3(X)=\frac{\displaystyle 1}{\displaystyle x_1^2+x_2^2+1}-1.1\exp\left[-\left(x_1^2+x_2^2\right)]\right]\\ \text{Domain } & -3\leq x_i\leq 3 \ \ , \ \ \ i=1,2 \end{cases}$$

\(\bullet\) The Pareto-optimal set of this \(2\)-dimensional three-objectives problem is:

$$\begin{eqnarray} f_1(\tau) &=& 0.5\tau + \sin(\tau)\\ f_3(\tau) &=& \frac{\displaystyle 1}{\displaystyle 1 + \tau} -1.1\exp(-\tau)\end{eqnarray}$$

where

\(\bullet\) \(\tau=x_1^2+x_2^2 \ \ , \ \ \ \tau \ \in \ [0,18]\)

\(\therefore\) If \(\tau\) is known, then the optimal value of \(f^{*}_2\) may become a candidate solution for the true Pareto-optimal set [1].

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 (2016). Unconstrained Multi-Objective Benchmark Functions Repository [https://www.al-roomi.org/benchmarks/multi-objective/unconstrained-list]. Halifax, Nova Scotia, Canada: Dalhousie University, Electrical and Computer Engineering.

Here is a BiBTeX citation as well:

@MISC{Al-Roomi2016,
author = {Ali R. Al-Roomi},
title = {{Unconstrained Multi-Objective Benchmark Functions Repository}},
year = {2016},
address = {Halifax, Nova Scotia, Canada},
institution = {Dalhousie University, Electrical and Computer Engineering},
url = {https://www.al-roomi.org/benchmarks/multi-objective/unconstrained-list}
}

III. References:

[1] K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms. Chichester, New York: John Wiley & Sons, 2001.
[2] Wikipedia, "Test Functions for Optimization — Wikipedia," 2016, [Online; Accessed Apr. 03, 2016]. [Online]. Available: https://en.wikipedia.org/wiki/Test_functions_for_optimization