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Why cannot we adapt Barvinok type counting techniques to general convex integer programs?

Decision problems in Integer Linear Programming have Lenstra type algorithms (https://www.math.leidenuniv.nl/~lenstrahw/PUBLICATIONS/1983i/art.pdf) have been generalized to convex integer program ...
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On convex quadratic programming clarification

We know convex quadratic programming is in $P$. Is it also in $P$ if the function of interest is only convex in the domain of interest?