FPT algorithm for mixed integer program It is known that every integer linear program parameterized by the number of variables is FPT (fixed parameter tractable). Is every mixed integer program parameterized by the number of integer variables also FPT?
The complexity of Lenstra's algorithm for mixed-integer programming in his paper runs as  $2^{O(n^3)}*poly(d, \phi)$  where there are $n$ integer variables, $d$ continuos variables, and $\phi$ is the binary encoding size of the problem.  This is not stated very explicitly in his paper, but the main thing to notice is that the complexity is primarily guided by the number of subproblems, which is $2^{O(n^3)}$.  Certain improvements allow a Lentra-type algorithm to have a complexity of $2^{O(n log(n))}$.  Other techniques allow for even n^n in terms of the number of integer variables.  

Note that the method of Lenstra is easily applied to convex mixed integer optimization provided certain separation oracles exist.  This dates back to a book of Gretchel, Lovasz, and Schrijver.  Other results on semi-algebraic convex optimization exist for the pure integer case as an FPT.