Given a positive definite matrix $A \in M_n$ and a vector $b \in \mathbb{R}^n$,
$$\begin{array}{ll} \text{minimize} & \frac{1}{2} x^T A x + b^T x\\ \text{subject to} & x \in \mathbb{Z}^n\end{array}$$
That is the program I am currently writing in Java. I succeeded in finding a solution which has real values. However, I hit a wall when it comes to finding integer solutions.
I considered genetic algorithms, checking every integer vector "around" real solution.
I would like to understand how to solve this problem. It doesn't matter if the solution is the optimal one, I wish it was an easy to understand one (and if it wasn't perhaps the worst case in terms of being optimal) so that I can get a better understanding of what is going on and keep researching in that area.
On $\mathbb{R}^n$, the minimum is reached for $x_0=-A^{-1}b$. We research a solution in the form $x_1=x_0+h\in\mathbb{Z}^n$.
EDIT. Correction. Then $F(x_1)-F(x_0)=1/2h^TAh$. It's more difficult! We are looking for some $h$ pretty much in the direction of the eigenvector of $A$ associated to the smallest eigenvalue of $A$.