# Question on non-linear parametric mixed integer program

I am trying to solve a mixed integer minimization problem, where there are a number of parameters, and there are products of parameters with variables appearing in the objective function. I assume that this counts as a non-linear program, due to the product of variables/parameters.

I tried searching for some kind of software or just some algorithm/methodology to solve a problem like this, but I haven't found anything. Is it even possible to get a solution as a function of the parameters?

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Your question does not have enough information to have an answer, nor is the question well formulated ("is it even possible"? You mean, in finite time? in the sense of mathematical logic? in less than an hour?) – Igor Rivin Nov 29 '11 at 19:38
I am sorry if it was not very clear, but I am obviously interested in doing this in finite time, since as I mentioned, I need an actual solution to this kind of problem. So what I need is a methodology and/or perhaps some related software tool. A negative type of answer is also helpful, as long as it contains some information. Or some hints on what kind of optimization problems involving parameters can be solved with well-known methods, since this seems to be an active research field. – cronopio Nov 29 '11 at 22:58
If I had to guess, it sounds like you're looking for a multiparametric MIP algorithm. M. Ierapetritou has published a few papers on this, but it is restricted to a limited class of MIPs. I don't know of any useful work on multiparametric programming for MINLPs -- we may be a decade away from that. If you are looking for multiparametric nonlinear programming for continuous NLPs, there is some work on that by E. Hale. She has a program called POPAK that does that. – Gilead Nov 30 '11 at 2:06
Can you provide more information, preferably in mathematical notation? What are the variables, constraints, objective, etc.? – Gilead Nov 30 '11 at 2:07
For fixed values of the parameters, it appears that you have an integer linear programming problem which can be solved by conventional branch and bound (or more sophisticated branch and cut) methods. There's lots of software available for that task. You haven't told us what you want to do with the parameters though. Do you need to know optimal solutions over a whole range of values of the parameters? – Brian Borchers Nov 30 '11 at 2:55