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

### Sotirios (view profile)

on 26 Dec 2013
Latest activity Commented on by Tarek Salem

on 18 Aug 2014

### Alan Weiss (view profile)

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?

### Alan Weiss (view profile)

on 26 Dec 2013

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

### Sotirios (view profile)

on 26 Dec 2013
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

### Alan Weiss (view profile)

on 27 Dec 2013
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.