(Note : I'm not sure about the tags, please re-tag this if you think you have the right tag).

I am optimising a certain function over a certain space (that i will describe), and to use non-constraint optimisations routines, i wanted to know if there is a mapping from $\mathbb{R}^{p}$, where $p$ is a constant to be determined, to the following space $S$:

Define first the index set $$\mathbb{N}_{n}^{d} = \left\{\mathbf k \in \mathbb{N}^d\, : \; k_i \le n \;\forall i \in 1,...,d\right\}$$

Define now the space $S$ as being the following restriction of $\mathbb{R}_{+}^{\mathbb{N}_{n}^{d}}$ :

$$y \in S \iff y \in \mathbb{R}_{+}^{\mathbb{N}_{n}^{d}} \text{ and }\forall i \in 1,...,n,\, \forall j \in 1,...,d,\, \sum_{\mathbf k \in \mathbb{N}_{n}^{d}} y_{\mathbf k} \mathbf 1_{k_j = i} =\text c_{i,j}$$

For some constants $\left(c_{i,j}\right)_{i \in 1,...,n,\, j \in 1,...,d}$ such that $\sum\limits_{i=1}^{n} c_{i,j} = C$ for all $j \in 1,...,d$

Note that this implies that $\sum\limits_{\mathbf k \in \mathbb{N}_{n}^{d}} y_{\mathbf k} = C$.

The insights is that, if $C = 1$, y are probabilities of some multivariate discrete distributions with marginal probabilities for the $j^{\text{th}}$ marginal $\left(c_{i,j}\right)_{i}$. $C$ is not one, so this measure is not a probability, but you could set $C = 1$ without any problem.

So the question is : Is there some way to map some power of $\mathbb{R}$ to this space $S$, that is surjective ? I dont care about injectivity by the way.