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Topic: Fmincon: nonlinear binary problem
Replies: 10   Last Post: Apr 12, 2013 7:07 AM

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Posts: 4
Registered: 4/10/13
Re: Fmincon: nonlinear binary problem
Posted: Apr 12, 2013 6:16 AM
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> > I was just thinking that all these 3 constraints can be written as one
> >
> > x_ik+x_jk>=2*z_ijk, right?
> >

> This condition does not exclude the case that z_ijk=0 although both x_ik and x_jk are equal to 1. So z_ijk is not equal to x_ik*x_jk. It depends on your objective function if z_ijk will automatically be chosen to be 1 if x_ik=1 and x_jk = 1.


> > "Torsten" wrote in message <kk60l8$l9i$1@newscl01ah.mathworks.com>...
> > > Because you substitute x_ik*x_jk = min(x_ik,x_jk) , I think you will need z_ijk:
> > > z_ijk <= x_ik
> > > z_ijk <= x_jk
> > > z_ijk >= x_ik + x_jk - 1

> >
> > Good point Torsten. Theoretically, I would need to index z over k but from other constraints I also know that there may be at most one k for which z_ijk=1 and I don't need to know the detailed information which k it is, but only the indication whether it happens at all. Would indexing z_ij be correct in that case? It's important to me to limit the size of the variables as well.

> I don't think that this is possible. Imagine there are indices k1 and k2 such that, for fixed i and j, x_ik1*x_jk1 = 1 and x_ik2*x_jk2 = 0. Then, if you only introduce z_ij, your problem becomes infeasible.

The original nonlinear objective was
obj=sum(i) sum (j) sum(k) x_ik*x_jk*p_ij

Maybe my explanation was not clear, I intend to replace sum(k) x_ik*x_jk=z_ij for every i and j. Would that be fine? It seems to me that all the previously proposed constraints should hold. Thank you for all your comments.


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