Hi Leonardo,

First you may check two sources of the optimal fronts of 3-D DTLZ instances:

> http://www.cs.cinvestav.mx/~emoobook/apendix-e/apendix-e.html

Hi Tsung-Che Chiang,

Thanks for your answer, but there is one issue regarding DTLZ7, how can compare the optimum solutions for two different algorithms?

is there any criteria just like the one in DTLZ1 or DTLZ2-DTLZ4 for the objective functions?

I am try to compare results between two algorithms to chose the better one

Kind regards

Jafar

Hi Jafar,

If you only need the reference optimal front of 3-D DTLZ7, you can download the files from Appendix E of Coello et al.' book or the resource page of jMatal software, as mentioned in my previous answer.

If you want to generate the optimal front by yourself, you need to check the original paper (whose link is also in my previous answer). Equation (6.25) on page 134 describes DTLZ7 function. In the text, it mentions that the Pareto-optimal solutions correspond to xM=0. Thus, g(xM) = 1. Accordingly, given the previous M-1 objective values f1, f2, ..., fM-1, we can get fM by 2*h.

Among all generated objective vectors, keep the non-dominated ones. Then, you have your own reference optimal front.

The code may look like this (3-D):

for (double x1=0; x1

Dear Dr. Tsung-Che Chiang,

Many thanks

Regards

Jafar

Hi Tsung-Che Chiang,

How  can  we generate the optimal fronts by ourselves for more than three objectives.

 5 - D data is attached. I don't know how it is generated.I'd like to generate the multidimensional data .Can you give me an example.

Hi Hwang Sun,

MOEAFramework provides the PF of DTLZ1-4 and DTLZ7 with 2, 3, 4, 6, and 8 objectives. You can download them here:

https://github.com/MOEAFramework/MOEAFramework/tree/master/pf

If you still need to generate PF of DTLZ1-4 by yourself, the paper of Li et al. [1] describes one method in details in Section IV-B. Just check equations (14) and (17). You may also use the PF generator included in nsga3cpp [2], which is an unofficial implementation of NSGA-III by me. Note that for 8 and more objectives, the number of Pareto optimal solutions could be very large. In this case, the 2-layer methoed mentioned in [3] is adopted. Li et al. [4] proposed another complex method to generate PF of DTLZ functions.

If you want to generate PF of other test functions, MOEAFramework provides a way to do so. Please check the discussions here [5].

I guess that you are generating PF of DTLZ and other functions since you want to measure algorithm performance by the inverted generational distance (IGD) indicator. In this regard, you may want to check Ishibuchi et al.' study [6] on an improved indicator called IGD+, which is weakly Pareto compliant (but IGD is not).

Finally, Ishibuchi et al. [7] indicated that DTLZ1-4 and WFG4-9 are similar functions with respect to the shape of PF and mentioned the necessity of more general test problems. Cheng et al. [8] proposed a new set of test functions with an open-source software platform PlatEMO [9], which may also be interesting to you.

Hope the above information is useful to you. :)

-------------------------------------

[1] Li et al., An evolutionary many-objective optimization algorithm based on dominance and decomposition, t-EC 2015, https://doi.org/10.1109/TEVC.2014.2373386

[2] nsga3cpp: http://web.ntnu.edu.tw/~tcchiang/publications/nsga3cpp/nsga3cpp.htm

[3] Deb and Jain, An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, Part I: solving problems with box constraints, tEC 2014. https://doi.org/10.1109/TEVC.2013.2281535

[4] Li et al., A method for distributing reference points uniformly along the Pareto front of DTLZ test functions in many-objective evolutionary optimization, ICIST 2015. https://doi.org/10.1109/ICIST.2015.7289031

[5] https://github.com/MOEAFramework/MOEAFramework/issues/49

[6] Ishibuchi et al, Modified distance calculation in generational distance and inverted generational distance, EMO 2015, http://doi.org/10.1007/978-3-319-15892-1_8

[7] Ishibuchi et al., Performance of decomposition-based many-objective algorithms stronly depends on Pareto front shapes, tEC 2017. https://doi.org/10.1109/TEVC.2016.2587749

[8] Cheng et al., A benchmark test suite for evolutionary many-objective optimization, CIS 2017. https://doi.org/10.1007/s40747-017-0039-7

[9] PlatEMO: http://bimk.ahu.edu.cn/index.php?s=/Index/Software/index.html

Dear Dr. Tsung-Che Chiang,

Many thanks

Kind regards 

Hwang Sun

Dr Tsung-Che Chiang 

Your information is precise. Thank you very much. It would help me to improve my NSGA-III C version.

Hi everyone. I obtained the pareto-optimal front for DTLZ standardized problems directly from Dr Tsung-Che Chiang unofficial code. The inverted dtlz1 was generated using the code that creates the reference points using the Das and Dennis approach. I also used the IGD metric to compare my results with authors' NSGA-III results.

Deb has used test problems with 5, 10 and 15 objectives in his paper. I am wondering if anyone has known pareto front for DTLZ* problems with 5, 10 and 15 D?

Hi,

I tested 5d and 10d with results similar to Dr Deb got.

You can see my channel on YouTube to see how the set of solutions evolved.

https://www.youtube.com/channel/UCMgWIVOV9bJtIfIxQOYhOuw

Regards,

I have Graph partition problem and I want to apply nsga2 or 3 on it but I cannot find the pure source code and I don't know the dltz problem

Tsung-Che Chiang

Hi Mr.Chiang!

1st of all lemme say thank to you for your complete explanation!

I've been a bit got confused about it!!!

1: what's the true PF??

2: IGD is used to qualify MOEAs so if we already knew about the true PF so what's the purpose of our new algorithms???

3: How did you get the true PF?? is that reliable?? did you used NSGA2 or 3???

Amirhosein Fardi

Hi Amirhossein

The true Pareto for benchmark optimization problem MUST be known in order to evaluate the quality of the solution obtained from different optimization algorithms. For example, IGD is simply the average distance of the Pareto front obtained from the algorithm from the true Pareto front. Without knowing the true PF we cannot calculate it.

In addition, when someone proposes an optimization test suite, they also present how to obtain the true PF. The true PF is not obtained from algorithms. Some problems are so hard that event the best algorithm we know cannot even approximate the general shape of the true PF.

The true PFs are calculated by plugging in specific values into the functions. As an example, the true PF for all of the problems of the DTLZ test suite is [0.5, 0.5, ...], where ... is any random vector between 0 and 1.

You can refer to the paper of Golden Eagle Optimizer and its multi-objective version proposed by me to see more about true PFs and the front obtained by MOGEO.

Original paper: https://www.sciencedirect.com/science/article/abs/pii/S0360835220307208

Preprint: Article Golden Eagle Optimizer: A nature-inspired metaheuristic algorithm

Source code and MATLAB toolbox: https://www.mathworks.com/matlabcentral/fileexchange/84430-golden-eagle-optimizer-toolbox

check this link, I hope it can help

https://github.com/BIMK/PlatEMO

Abdolkarim Mohammadi-Balani

Hi Abdolkarim

1st of all an special tnx to you for your complete answer to my Q.

That was too help full tome

I tried to learn it so I read some articles abt test function by Zitzler but I'm still a bit confused abt how to construct a proper test function!!

As if u know there are 3 approaches for generating a scalable test function and one the most famous of them is buttom-Up approach in which before defining functions we're gonna set a true pareto front.

For exp for M objectives, this means designing a parametric surface involving (M − 1) variables!

After that the search space and the variable space is gonna be based on the mathematical equation of true pareto front defined above! So what's the criteria for such an equation of the desired pareto front???

Also you know that in DTLZ1, the last k variables are represented in Xm and it's been said that "The Pareto-optimal solution corresponds to xi = 0.5 (for all xi ∈ Xm) and the objective function values lie on the linear hyper-plane", so why we set variables in Xm to 0.5??

Abdolkarim Mohammadi-Balani

And also could you possibly tell me how I can achieve the true pareto fronts for DTLZ1 of 10 obj function!!

is its table downloadable??