Given $y_1,\cdots,y_N\in \mathbb{R}$, find $\mathbf{x}_1,\cdots,\mathbf{x}_N\in \mathbb{R}^K$, subject to some linear constraints.

Question

Assume that $y_1,\cdots,y_N\in \mathbb{R}$ are known.

Now I want to find a set of $\mathbf{x}_1,\cdots,\mathbf{x}_N\in \mathbb{R}^K$, such that $y_n=\mathbf{x}_n^T\mathbf{a}$, $\sum_{n=1}^N\mathbf{x}_n=\mathbf{b}$ and $0\le\mathbf{x}_n\le1$.

More clearly:

$Find\quad \mathbf{x}_1,\cdots,\mathbf{x}_N\in \mathbb{R}^K\\ s.t.\quad y_n=\mathbf{x}_n^T\mathbf{a}\\ \quad\quad \sum_{n=1}^N\mathbf{x}_n=\mathbf{b}\\ \quad\quad 0\le\mathbf{x}_n\le1 $

Any suggestion is appreciated!