How to extract the non-dominated solutions (Pareto front)

Question

I have tried so many times to write the code of extracting the non-dominated (or dominated)

solutions for two objective functions using MATLAB.

I have two simple objective functions:

J1=x.^2

J2=(x-2).^2

and I have a range for x values, say from -5 to 5 and there are, for example, 100 solutions to be

generated randomly within the range specified.

I want to extract the non-dominated solutions from these solutions.

I have no problem of all above operations. What I have done so far is:

   % generating 100 solutions randomly between -5 and 5:

   x=-5+10*rand(100,1);

   % calculate both objective functions, J1 and J2 at each solution:

   J1=x.^2;

   J2=(x-2).^2;

Now, I faced the problem of how to translate the concept to a written code.

I know the concept of how to extract the non-dominated solutions and Pareto front.

I can do it manually but this will take very long time.

I tried using if statements but the results were not accurate.

I think it is better to extract the indices of the dominated solutions and then remove them from

the main vector x to get the non-dominated solutions.

Thanks in advance

Answer 1

do you mind files from FEX? This one works perfectly: "Pareto Front" by Yi Cao

giving you the indices of x of dominating solutions.

Then you just have to use it like this:

x=-5+10*rand(100,1);
J1=x.^2;
J2=(x-2).^2;

idx = paretofront([J1,J2]);
xdi = ~ismember(idx,1:numel(x));

figure(1)
hold on
scatter(J1,J2,10,'red');
scatter(J1(idx),J2(idx),50,'blue');
scatter(J1(xdi),J2(xdi),50,'green');
hold off

legend('all solutions','dominating solutions','non dominating solutions')

leads to:

which is exactly how it is supposed to look like. Otherwise you need to clarify your question.