Consider the Bregman divergence induced by negative entropy function $$ D_h(x||y) = -\sum x_i + \sum y_i +\sum x_i\ln \frac{x_i}{y_i} $$
Given a constrained simplex $K_\alpha = [x\in [\alpha,1]^d | \sum x_i = 1]$ with $\alpha\in [0,1/d]$. What is the Bregman projection of a non-negative vector $y$ to $K_\alpha$? In other words, given $y\in R_+^d$, what is $y^+$ defined as $$ y^+ = \arg \min_{x\in K_\alpha} D_h(x||y) $$ ?
It's known that when $\alpha=0$, $y^+ = \frac{y}{||y||_1}$. I am wondering if there is a closed form for $y^+$ for general $\alpha$.
Thanks in advance!