Given a basis for a full-rank lattice $\mathcal{L} \subset \mathbb{R}^n$ I want to find a vector with totally positive entries, in other words an element belonging to $\mathcal{L} \cap Q$ where $Q$ is the "quadrant" $\{(x\_1, ..., x\_n) \in \mathbb{R}^n | x\_i \gt 0, 1 \leq i \leq n\}$.


The only construction I can come up with uses doubling and adding of basis vectors. Is there a simpler way (at least intuitively if not computationally)?


It's also important to me that this vector is not too large and I worry a double and add approach will be suboptimal. Are there any variants of shortest vector problem (SVP) algorithms which can find a short vector in a "quadrant" like this?



How can I construct an element in a particular "quadrant" of a lattice (preferably short)?
Since I haven't had a response I'll add my temporary solution. You can use Babai round off with suitably short vectors in the positive orthant to find vectors in the lattice that may be in that orthant. Its not optimal but it is fast.


