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I am new to the optimization problem of Quadratic programming. In equation 8 of the following paper: here , there is an equation:

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The authors state that this is an 'Integer Quadratic Programming (IQP)' formula.

Alternatively, in another website: here , there is the following equation which is described as a 'Mixed Integer Quadratic Programming (MIQP)' formulation:

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From my perspective, both of the equations shown above are similar, with the only difference being that the MIQP formula has '1/2' included in it.

1) I am looking for an explanation on the differences between the IQP and MIQP

2) In addition, I am interested to apply quadratic programming to the assignment problem, thus, looking for any insight into which should be used (i.e., IQP vs. MIQP) and when.

user121
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Integer Quadratic Programming (IQP) implies there are no continuous variables in the model: all variables are discrete. Mixed Integer Quadratic Programming (MIQP) allows both discrete and continuous variables. If your model only has discrete variables it is both an MIQP and an IQP. All popular solvers are of the MIQP type, so I tend to use MIQP even if I don't have continuous variables. IQP as model type is not often used. I don't think this is really something to worry about.

Erwin Kalvelagen
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  • Erwin, so would you say that if MIQP was used in the paper (with Equation 8 posted above) that it would give the exact same solution as IQP ? Also would you be able to point me to a simple comparison of MIQP vs IQP in Python or R. – user121 Mar 15 '19 at 20:37
  • Erwin, I do not have enough experience in QP to know what's out there in terms solvers. On another note, not sure if you know any source off-hand that you can point me to, but I am also looking for an example of the assignment problem using MIQP. Cheers. – user121 Mar 16 '19 at 00:08
  • As said, this is largely a non-issue. There is no reason not to use a MIQP solver for pure integer problems. There are many different assignment problems, so this is difficult to answer. – Erwin Kalvelagen Mar 16 '19 at 13:00