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Topic: Lower bound of a non-convex program through the dual problem in Matlab
Replies: 5   Last Post: Jan 23, 2013 3:59 AM

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Johan Lofberg

Posts: 48
Registered: 11/6/06
Re: Lower bound of a non-convex program through the dual problem in Matlab
Posted: Jan 23, 2013 3:59 AM
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With, YALMIP, you can solve small problems to global optimality. By tweaking the options, you can tell it to give up early and return a valid lower bound (on the minimization objective)

sdpvar x
Constraints = [0<= x <= 50];
Objective = exp(-x).*cos(2*pi*x);
options = sdpsettings('solver','bmibnb');
% Solved to global optimality after 5 calls to fmincon
% (and a bunch of LP calls for bound propagation)
solvesdp(Constraints, Objective,options);

% terminate early and extract lower bound
options = sdpsettings('solver','bmibnb','savesolveroutput',1,'bmibnb.maxiter',1);
solvesdp(Constraints, Objective,options);

Might be an overly complex strategy though, in the sense that it implements a branch-and-bound strategy (you would thus have a b&b code compute lower bounds in your b&b code...)

"Nazmul Islam" wrote in message <kdmpr0$k31$>...
> Bruno,
> That's a good observation! :) I am actually maximizing an integer non-concave nonlinear program :S. I am using branch and bound to solve the problem, i.e.:
> 1. I make some decision on the integer variables.
> 2. For each decision variable, I try to get an upper and lower bound of the problem.
> 3. I compare the bounds of different branches (integer variables) and remove the worse ones.
> 4. After some iterations, I will stop, pick the best branch (integer variable) and hopefully know how far I am from the optimum (using the best upper bound).
> Any feasible solution of a non-concave program can be treated as its lower bound. But, I need some ways to calculate a good upper bound (not +Inf :) ). I am looking for some software that can help me.
> Thanks,
> Nazmul
> "Bruno Luong" <b.luong@fogale.findmycountry> wrote in message <kdmivl$nth$>...

> > "Nazmul Islam" wrote in message <kdmigl$mcj$>...
> >

> > >
> > > Unfortunately, my solution is getting stuck at a local minimum. Please let me know if I can get the lower bounds of this problem through the interior point method.

> >
> > -Inf is lower-bound. Of course it is useless answer, nevertheless it is a correct answer (You never explain why you need a lower-bound for).
> >
> > Bruno

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