Given $w \in \mathbb R^{M}$ and symmetric positive semidefinite matrix $B \in \mathbb R^{M \times M}$, find $x \in \mathbb R^{M}$ such that
$$\begin{array}{ll} \text{minimize} & (x-w)^{T}B(x-w)\\ \text{subject to} & \|x\|_{1} \leq 1\end{array}$$
I was thinking if it's possible to break the problem into two parts, thus if I already can perform projection onto the $\ell_1$ unit ball efficiently, then would it be possible to solve for unconstrained problem of minimizing $(x-w)^{T}B(x-w)$, and if the $\ell_1$ norm of $x$ turns out to be greater than $1$, then I can perform the efficient projection of $x$ onto the $\ell_1$ unit ball, via
$$\arg\min {1 \over 2} \|x - y\|_2^2$$ given $$\|y\|_{1} \leq 1$$