Let $K$ be the closed convex set of $\mathbb{R}^N$ defined as the intersection of:
- the first orthant $\mathbb{R^+}^N$
- the hyperplane $\text{Ker}(\phi)$ where $\phi(x)= 1-\langle x,n\rangle$ with $n$ a unitary vector
We can express the projection onto $\text{Ker}(\phi)$ as:
$$ p:x \mapsto x + \left(1-\langle x,n\rangle\right) n$$
Can we similarly express the projection onto $K$? For instance, in the simpler case where $\forall i, n_i=\frac{1}{\sqrt{N}}$.
Edit: The following is wrong, I keep it so that the comment of joriki can be understood.
Composing projections onto the orthant and the hyperplane might work, since the projection onto the hyperplane maps the orthant to K.
Best Answer
As explained in an answer to another question, the article Projection Onto A Simplex by Yunmei Chen and Xiaojing Ye provides a relevant answer. There is even a piece of code in Matlab.