Decreasing Function Projected onto Simplex

Question

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$, defined as $f(x) := a x + b$, where $a<0$ and $b \in \mathbb{R}_{\leq 0}^{n}$.

Note that $f$ is decreasing: $$ x \geq y \Longrightarrow f(x) \leq f(y). $$

Consider the set $$ S := \{ x \in \mathbb{R}^n \mid x\geq 0, \ 1^\top x = 1 \} $$ and let $P_S: \mathbb{R}^n \rightarrow S$ be the Euclidean projection onto S ($P_S(x) := \arg \min_{y \in S} \left\|x-y \right\|$).

It holds that $P_S( f(\cdot) )$ is not decreasing.

Say if $-P_S( f(\cdot) )$ is monotone, at least on the positive orthant: $$ \left( P_S(f(x)) - P_S(f(y)) \right)^\top \left( x-y\right) \leq 0 $$ for all $x,y \geq 0$.

Answer 1

The elements of $S$ are not comparable to each other in the pointwise order of $\mathbb{R}^n$. In other words, if $s,t\in S$ and $s\geqslant t$, then $s=t$. This means that you might as well write the property you want as $$ x\geq y\geq 0\implies P_S(f(x))=P_S(f(y)).$$ This implies that the projection of a whole cone would go to a point, which should never hold.