Multi objective optimization with intlinprog

Hello,

My problem is a linear optimization problem with integer values, so I followed from http://www.mathworks.com/help/optim/ug/intlinprog.html and was able to make it work just I way I want it. Now I have two optimization functions and the remaining problem is the same. Multi objective optimization works fine on http://www.mathworks.com/help/gads/examples/performing-a-multiobjective-optimization-using-the-genetic-algorithm.html. Hoever, I need to do the ssamre with integer values.

Is this possible using intlinprog?

Thanks

Answer 1

[Matt J]

One possibility is to apply your own manual scalarization scheme, e.g.,

     weights=linspace(0,1,10);

     for i=1:length(weights)

        alpha=weight(i);

        f=alpha*Objective1+(1-alpha)*Objective2;

        [X(i,:), Fval(i,:)]= intlinprog(f,...);

      end

[Matt J]

If you have the Parallel Computing Toolbox, you can use a parfor-loop instead to speed things up.

[Victor Hugo Cantú]

I'm trying to solve a linear programming with two objectives. For one objective the code works pretty good using "linprog".

For two objectives, I already coded the weighted sum approach, similar to the one you proposed, and it works well. However, I would like to use another scalarization approach, i.e., Tchebycheff or AASF. For nonlinear functions, "fmincon" works well, but with "linprog" I'm having troubles to code it, mainly because the objective function is represented as the coefficients of the function. Is there any other way to send "f" to "linprog" instead of a vector of coefficients ?

Here I put an example of how Tcehebycheff works for "fmincon":

function f = fun(x,z,A)
f1 = sum(x.^2,2);
f2 = 3*x(:,1) + 2*x(:,2) - x(:,3)/3 + 0.01*(x(:,4) - x(:,5)).^3;
f = max((f1 - z(1))*A(1), (f2 - z(2))*A(2));
% f = f1*A(1) + f2*A(2); % for weighted sum approach
end

where z is the ideal point, a vector of objectives in which each element is the best solution found for every objective up to now (for being used for evolutionary algorithms). Even if one does not use the z, I see difficult how to transfert the f to "linprog":

f = max(f1*A(1), f2*A(2));

It is to note again, that f is a vector of coefficients.

Thank you in advance.

[Matt J]

See my answer to your post

https://www.mathworks.com/matlabcentral/answers/482215-mutliobjective-optimization-with-linprogr

Note, the way you use fmincon is not to be trusted because it is applied to a non-differentiable objective.

Answer 2

[Alberto Chavez]

This may be a little bit late, but have you tried the constraint method for your multi-objective optimization?, If you indicate your first function as your new constraint (for second objective) and the optimized first value as your new equality/inequality, that may serve to calculate new objectives while maintaining the optimal on the first functions. You may say that your "weights" is the order in which you solve each objective.