Let $N_i=\{0,1,\dots,\bar{n}_i\}$ and define $N=N_1\times \dots \times N_I$. I have a minimization problem of the form $$ \min_{n\in N} \sum_i A_i n_i +\sum_i \sum_{j\neq i} B_{ij} (n_i-n_j)^2 $$ where $A_i\geq 0$ and $B_{ij}\geq 0$ for all $i,j$. This problem can be written as a convex quadratic integer program.
What are the standard algorithms to solve these problems numerically? Are they guaranteed to find a global solution? How fast are they?
Typically, you'd use a branch and bound algorithm to solve this problem and obtain a globally optimal solution if the problem isn't unbounded. This does require worst-case exponential time.
Several commercial LP/QP packages have support for solving these MIQP problems.