Is it possible to generate *surface* pareto front for 3 objective functions and plot it?

Is it possible to generate surface pareto front for 3 objective functions and plot it? My actual question is that when we have two objectives, the Pareto front is a line in 2D space and when we have three objectives the Pareto front is a surface in 3d space. How we can obtain that surface?

Answer 1

[Alan Weiss]

Sure.

function f = simple_multiobj2(x,a,b,c)
x = x(:); a = a(:); b = b(:); c = c(:); % all column vectors
f(1) = sqrt(1+norm(x-a)^2);
f(2) = 0.5*sqrt(1+norm(x-b)^2) + 2;
f(3) = 0.25*sqrt(1+norm(x-c)^2) - 4;

Then run this:

a = zeros(2,1);
b = [2;1];
c = [3;-.5];
fun = @(u)simple_multiobj2(u,a,b,c);
[x,f,ef] = gamultiobj(fun,2)
scatter3(f(:,1),f(:,2),f(:,3),'k.');

If you want to plot an interpolated surface or mesh, use scatteredInterpolant.

Alan Weiss

MATLAB mathematical toolbox documentation

[Sotirios]

Thank you very much for your answer. Could you please tell me how to use the scatteredinterpolant in the example you gave to generate a surface plot? My actual question is that when we have two objectives, the Pareto front is a line in 2D space and when we have three objectives the Pareto front is a surface in 3d space. How we can obtain that surface?

[Alan Weiss]

F = scatteredInterpolant(f(:,1),f(:,2),f(:,3),'linear','none');
sgr = min(f(:,1)):.01:max(f(:,1));
ygr = min(f(:,2)):.01:max(f(:,2));
[XX,YY] = meshgrid(sgr,ygr);
ZZ = F(XX,YY);
surf(XX,YY,ZZ,'LineStyle','none')

The 'none' argument in scatteredInterpolant removes any extrapolated points, so you just se the true extent of the calculated surface, assuming that linear interpolation shows the truth.

If this sufficiently answers your question, please accept the answer.

Alan Weiss

MATLAB mathematical toolbox documentation

[Tarek Salem]

please i i need to plot 4D function, because the algorithm optimize 4 functions at a time