Graphing a Multi-Objective Optimization problem

I am attempting to plot the pareto graph of a bi-objective problem(self learning) which goes: ie maximize

f1 = x1

f2 = x2

such that

𝑥1^2 + 𝑥2^2 - 1 ≤ 0

2𝑥1 + 𝑥2 - 2 ≤ 0

I have confirmed the correct sketch of the graph theoretically but I have spent days trying to read up on various ways to use matlab to plot it and I never could do it since my handle of matlab is mediocre at best. My final attempt was using gamultiobj in the toolbox but the graph is totally wrong (followed video on youtube). For the toolbox app, I used two scripts:

function Output = multi_objective_function(Input)

x1 = Input(1);
x2 = Input(2);

f1 = x1;
f2 = x2;

Output = [f1 f2];

and

function [C Ceq] = nonlinear_constraints(Input)

x1 = Input(1);
x2 = Input(2);

C = x1^2 + x2^2 - 1;

Ceq = []; 

and entered A = [2 1] and beq = [2] in the linear inequality constraints section of the toolbox app window. But the graph came out totally different than expected. I tried several variants of the scripts including negating f1 and f2 (ie f1 = -x1 & f2 = -x2) in case the app only did minimizations. So

(1) any sort of help on the best approach will be appreciated and

(2) For the toolbox, did I do it right? Thanks and happy holidays.

Answer 1

[Alan Weiss]

Using your functions (edited versions below), I called gamultiobj as follows:

A = [2 1];
b = 2;
options = optimoptions('gamultiobj','PlotFcn','gaplotpareto','PopulationSize',200);
[x,fval] = gamultiobj(@multi_objective_function,2,A,b,[],[],[-1 -1],[1 1],@nonlinear_constraints,options);

Then, to plot the resulting points:

figure
plot(x(:,1),x(:,2),'ko')

Does that look like what you expect?

Here are the functions I used:

function Output = multi_objective_function(Input)

x1 = Input(1);
x2 = Input(2);

f1 = -x1;
f2 = -x2;

Output = [f1 f2]; % Coould have reduced to Output = -Input;
end

function [C,Ceq] = nonlinear_constraints(Input)

x1 = Input(1);
x2 = Input(2);

C = x1^2 + x2^2 - 1;

Ceq = []; 
end

Alan Weiss

MATLAB mathematical toolbox documentation

Answer 2

[John Evans]

Thank you very much. Stay safe.