Originally posted by: SystemAdmin

[miso said:]

Dear All,

I am trying to define an optimization problem containing a quadratic objective function as well as a number of quadratic constraints, which can be described as followed.

objective function:
minimize: dx * dy; dx and dy being positive integer values

subject to:

…linear constraints and the following quadratic constraints

lower boundary <=  (dx*c) *(dy*c)  <= upper boundary; c being a constant allowing scaling<br />
Is there a way to setup the quadratic constraint allowing Q to remain positive semi definite and therefore to solve the problem?
I am using Opl 6.1.

Thanks for your help. Best Regards.

#DecisionOptimization
#MathematicalProgramming-General

Originally posted by: SystemAdmin

[miso said:]

While searching through the forum I realized that, since the decision variables can be integers, such problem could be optimized by using CP. I haven't tried CP programming yet, though.

Nevertheless I am still interested to know whether such statement could be solved via quadratic programming.

Thank You and Regards

#DecisionOptimization
#MathematicalProgramming-General

Originally posted by: SystemAdmin

[jfk said:]

Hello,
1. if your variables are all integer, then I think, basically any quadratic programming model can be solved by CP. The only thing you have to do is put "using CP;" at the beginning of your code and run the respective runconfig.

2. while it seems to be a bit unusual to adjust the data to solve a given problem - like making the coat to match the button :-) -, making a matrix positive semidefinite, I think,  is an opti problem on its own:
there are several characterization criterion for M to be positive semidefinite, see e.g. http://en.wikipedia.org/wiki/Positive_semidefinite_matrix
You have to choose one of the criteria (e.g. the Syslvester criterion is very straightforward but computationally expensive). Then you have to take the given criterion (basically it will become a set of constraints) with variables for some or all of the elements in M - I suppose you want to be as close to the original matrix as possible -and solve the opti prb with minimizing the objective sum(i,j)(m[sub]i,j[/sub]-mvar[sub]i,j[/sub])[sup]2[/sup] where m[sub]i,j[/sub] are the original elements of the matrix M and mvar[sub]i,j[/sub] are the variables. if the mvar[sub]i,j[/sub] can be real than it is an NLP. But again if you choose for the variables of M to be integer then you can solve it by CP.

I hope it helps
cheers
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#MathematicalProgramming-General

Originally posted by: SystemAdmin

[miso said:]

Dear jfk

Thank you for the quick reply. I am currently trying to understand and test using cp; I hope I can scale the variables in order to make the integer solutions a little more precise. This might be the way to get around the quadratic problem.

Regarding solving the statement via quadratic programming I will probably need to spend some more time to understand the mathematical background better. I actually did not mean to change the data, I was basically hoping for a hint whether the constraint definition could be transformed or stated differently, which I did.

I will look into it; I really appreciate the help. Thank You. Best Regards

#DecisionOptimization
#MathematicalProgramming-General