Originally posted by: SlavaMedvedev

Hi!
I get CPLEX exception when I solve some of MIQCP problems.
Message:
CPLEX(default) cannot extract model: IloAlgorithm cannot extract extractable

Example:
dvar boolean x;
maximize
x;
subject to {
x*x == 1;
}

However when I modify this example to equivalent problem...
ExampleEx:
dvar boolean x;
maximize
x;
subject to {
x*x >= 1;
x*x <= 1;
}

... everything works fine.

Any help? Thanks!
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Originally posted by: SystemAdmin

The underlying C method that is called by Concert for quadratic constraints is CPXaddqconstr(). This allows only 'L' and 'G' constraints, because quadratic equations will immediately make the problem non-convex, at least in theory.
In your example, in the second case you are just lucky that the presolver is able to discard all constraints and you end up with a convex problem, hence CPLEX is able to solve it.
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Originally posted by: SlavaMedvedev

Thanks Tobias!

I just expected Concert to return something like "Q in constraint is not positive semi-definite" as it does for non-convex objective function minimization. Instead I receive internal exception and don't know what to do.

Now I have two examples again regarding QP problems.

Example1:
dvar boolean xhttp://1..5;
maximize
x[1]*x[2] + x[3]*x[4] + x[5]*x[5];
subject to {
x[1] + x[2] + x[3] + x[4] + x[5] <= 3;
}

Works fine!

Example2:
dvar float+ xhttp://1..5;
maximize
x[1]*x[2] + x[3]*x[4] + x[5]*x[5];
subject to {
x[1] + x[2] + x[3] + x[4] + x[5] <= 3;
forall (i in 1..5) x[i] <= 1;
}

Error 5002: Q in objective is not positive semi-definite.

Which is strange because I believe the second case is the root relaxation for the first one. Probably the possible reason is that MIP performs "Repairing indefinite Q in the objective."
Does that mean the MIQP supports more widespread space of possible Q in objective???
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Originally posted by: SystemAdmin

With binary variables you can use a simple trick to make a non-PSD objective function positive semi-definite: just add the valid equation x^2 - x = 0 to the objective function to make the diagonal of Q large enough to make Q psd. With continuous variables, this trick is no longer possible as the equation does no longer hold. Therefore, CPLEX can solve your model with binary variables, but is not able to solve it with continuous variables.
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Originally posted by: SlavaMedvedev

Thanks a lot!

As I understand:

If I use binary variables only I don't need to concern myself with Q in objective. CPLEX will automatically repair indefinite Q.

Is that true?
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Originally posted by: SystemAdmin

Yes, it is. But you have to note that the trick with exploiting x = x^2 can really degrade the tightness of your relaxation and thus the performance of the solving process.
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Originally posted by: SlavaMedvedev

You helped me very much because I have not found such information in CPLEX help. Thanks!
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