Does anyone know a reference to the answer if unconstrained integer convex optimization problem (i.e. $\min_{x\in \mathbb{Z}^N} F(x)$, $F$ is convex and $N$ is NOT fixed) is NP-hard?

Thank you in advance.

Well, thanks to your critical comments I've found the answer to the question, and here it is. Let $F$ be a piecewise linear convex function, i.e. $F(x)=\max_{j\in J} \sum_{i=1}^N a_i^j x_i + a_0^j$, then $\min_{x\in \mathbb{Z}^N} F(x) = \min_{x\in \mathbb{Z}^N} \max_{j\in J}\{\sum_{i=1}^N a_i^j x_i + a_0^j\} = \min_{x\in \mathbb{Z}^N} \min_{f\in \mathbb{R}} (f, s.t. f\geq \sum_{i=1}^N a_i^j x_i + a_0^j, \forall j\in J)$, the last optimization problem is a mixed integer linear program, which is known to be NP-Hard.

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