I have a vector $v \in \mathbb{N}^k$ and a set of vectors $R \subset \mathbb{N}^k$, with $k \ll \left\vert R \right\vert $.
I would like to find a way to obtain all the possible bases of $\mathbb{N}^k$ taking $k$ elements from $R$.
The ultimate goal is to find a decomposition of $v$ over a subset $S = \left\{ r_1, r_2, \ldots, r_q\right\} \subset R $ that will minimize a given cost function $f$.
Unfortunately the search space is really large ($\left\vert R \right\vert > 10^5$ even for basic problems), so I cannot just try to brute force a solution.
Is there any algorithm or theorem related to my problem that I could exploit?
UPDATE
It has been pointed out that $\mathbb{N}^k$ is not actually a vector space. I guess that it would be more correct to frame the problem in $\mathbb{Z}^k$, although all the components of all the vectors in my problem must be non-negative. I guess that a more correct statement is that $v \in (\mathbb{Z^+} \cup \left\{0\right\})^k$.
I forgot to specify an important condition: all the vectors in $R$ are such that the sum of their components does not exceed a given constant $T$. In fact, $R$ is defined as
$R = \left\{r \in \mathbb{N}^k \mid \sum r_i \le T \right\}$
When I say "decomposition of $v$ ", I mean that I would like to find a set of vectors $S = \left\{r_1, r_2, \ldots, r_n\right\} \subset R \subset \mathbb{N}^k$ such that $v$ is a linear combination of the elements of $S$. I am also assuming that $ 0 \in \mathbb{N}$. In formula:
$v = \sum x_i \cdot r_i$
where $x_i \in \mathbb{R}$ (truth be told, I am especially interested in the solutions where $x_i \in \mathbb{N}$, although those are simply better solutions than the non-integer ones).
The cost function is still under development and thus I don't have a precise formulation to share right now. Roughly speaking, it will be proportional to the cardinality of $S$, inversely proportional to the norm of each vector and inversely proportional to the number of non-zero components of each vector. Therefore, something essentially like this:
$ C(S) \propto \frac{\vert S \vert}{\sum \Vert s_i \Vert \cdot \sum V(s_i)}$
where $V \left(s\right)$ counts the non-zero components of $s$.
