MATLAB: Is there a work around to use gamultiobj with linear constraints and binary variables

binary optimizationgamultiobjlinear constraintsMATLABmultiobjective optimization

Hi,

I'm trying to minimize a linear and several quadratic functions with gamultiobj like the following:

 y(8)=    ((Befaehiger(:,2).*VF2(8,:)') .* x')'*(1+ (VF1(8,:)'.*x')' * FCM*0.1)'-0.23;
 y(9)=    x*Befaehiger(:,1);

Befaehiger,FCM, VF1 and VF2 are arrays. x are the binary variables.

My optimization problem has also linear inequality constraints.

Unfortunately my variables are binary. So when using 'PopulationType','bitstring' constraints are not supported. Setting variables as integers (intcon), combined with using bounds 0 and 1 is, also not possible with gamultiobj.

Does someone know a way to use gamultiobj (or another way) to minimize these functions with binary variables?

Thank you for reading

Steffen

Best Answer

I was thinking if the problem was not too large we could directly compute all 2^n possibilities. Obviously this is not something we would want to do. So, I looked into Walter's suggestion and was able to implement a smaller example.

The implementation I created is not guaranteed to work well with large problems as there is the possibility of having to re-sample data many times to satisfy the constraints. You would likely need to implement some specific re-sampling which is more probable to produce constraint satisfying points.

Please see the attached files. The main file is PopExample.m.

Up-voting Walter for his suggestion.

Related Solutions

MATLAB: I want to solve a optimization problem with binary variables, linear and nonlinear constraints using the genetic algorithm from the optimization toolbox. Is this possible

I suggest that you try making a linearly feasible initial population using intlinprog. To get a well-distributed population, use a different random objective function vector f for each population member:

 f = randn(n,1); % n is the total number of problem variables

Use this initial population, and the population should remain feasible with respect to bounds and linear constraints as iterations progress.

I suggest that you use a very large population and be prepared to wait. Integer programming in ga is not recommended for more than about 100 variables, so it is not clear whether this approach will work or not.

It seems to me that your nonlinear constraints are quadratic. If that is true, perhaps you could use an integer programming approach along the lines of this example. But I did not examine your constraints in detail, I might have missed something.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

MATLAB: GA violates the linear inequality constraints

You have expressed your constraints as nonlinear inequality constraints, not as linear inequality constraints. If the constraints are truly linear (they don't look linear to me), then you might have better luck expressing them in the format of linear inequality constraints.

You have a relatively large number of binary variables, and a small population. I would not expect ga to perform well on this kind of problem. And I do not understand why you set EliteCount to 1.

It seems to me that you have 23 binary variables. That is only 8388608 possibilities. Perhaps exhaustive search would work.