How can we calculate the true PF of the DTLZ test function!?

02 July 2021 7 3K Report

In optimization of sphere we already knows that the optimal point is the Origin and this is achieved by analytical optimization procedures and as we run Many/Multi Evolutionary Algorithms, the results are compared with the true PF.

Now I wanna know how to compute the true PF of normalized test Function such as DTLZ, DTZ and etc!!!

Qamar Ul Islam

Dear Amirhosein Fardi

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

Amirhosein Fardi

Qamar Ul Islam

Hi dear brother

tnx so much for this complete reply

I'm gonna read it and ask you again if necessary

with all resprect

Qamar Ul Islam

Your very welcome Amirhosein Fardi

Stay Happy Stay Healthy.

Amirhosein Fardi

I myself tried to understand it by plotting a 2D MOP2 test function with 5 variables in which I evaluated about 5000000 random input vectors and plotted the 1st and 2nd objectives in a 2D plane!

The result is clearly telling me that the true PF of the test function can be achieved so easily!

if you copy the code below, the result is obvious and helpful:

clc

clear

n= 5000000;

z=zeros(2,n);

for i=1:n

x= unifrnd(-1,1,[1 4 ]);

z(:,i)= MOP2(x);

end

plot(z(1,:),z(2,:))

xlabel('1st Objective');

ylabel('2nd Objective');

grid on;

Amirhosein Fardi

Qamar Ul Islam

Hi bro

I've been a bit get confused about DTLZ1 test functions!

I've coded the m-Objective DTLZ1 test function in MATLAB as follows:

function y = DTLZ1(x,m) % m is The Number Objectives

n = numel(x);

k = n-m+1;

g = 100*(k +sum((x(end-k+1:end)-0.5).^2 - cos(20*pi*(x(end-k+1:end)-0.5)) ) );

y = zeros(m,1);

for i=1:m

if i==1

z= 0.5*prod(x(1:m-1))*(1+g);

elseif i==m

z= 0.5*(1-x(1))*(1+g);

else

z= 0.5* prod(x(1:m-2-(i-2)))*(1-x(m-1-(i-1)))*(1+g);

end

y(i) = z;

end

I have some question about them:

Are these m functions in DTLZ1, conflicting with each other? If yes, I found the decision vector that can make all the m functions zero. can you justify this?

Why the True PF of this functions is obtained by setting the values of the first M-1 variables randomly and set the remaining k variables by 0.5? You know that this procedure leads to the line y = -x+0.5 in 2D and a plane in 3D, while as mentioned above, I found some decision vectors in 2D that make the Objective functions ZERO and this point is obviously dominates the points on the true PF.

As mentioned in NSGA3, the predefined well-distributed points called Reference Points are generated in order to maintain the diversity of the search space and to avoid the local convergence in the interval of [0,1]. Also the Cost functions are Normalized by the means of scalarization and are mapped to the interval of [0,1] to make them comparable with each other. Next the Euclidean distance between the Normalized Cost function of each member and all the Reference Points is calculated and the Ref. Point with minimum distance is associated to the member. It may be some Ref. Points that aren't associated to the population members. So now for the IGD performance indicator, the Euclidean distance between two set of solution should be calculated! These two set can be the True PF and the obtained PF by the algorithm and also can be the generated Ref. points and the Normalized Cost functions. I don't know which 2 set to be chosen!!!!

please tell me if I'm making mistake in understanding the concept of the Ref. Points and Normalized Cost function and their application in NSGA3.

Also I'm keen on talking with you directly in social medias if you want and have a time.

Kind regards

Amirhosein Fardi

Today I was reading the article by Yifan Li et. al that was too helpful to me for understanding the concept of DTLZ test function.

I attached it below

Amirhosein Fardi

I have plotted each of the the 5 Objective DTLZ1 test functions separately and I was get shocked by the result in which in some intervals they are not conflicting with each other and now I have Q that how they can be used as an conflicting functions in dealing with many Objective Optimization Problems???

Each color illustrates one of the objective in the attached picture

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