I am looking to see if it possible to optimize the following objective:
Minimize $(\| Ax - b \|_1) + (\| Ax - c \|_1)$ wrt to $x$ such that $\| x \|_1 = 1$ and all elements of $x$ are nonnegative. $A$ is a matrix, and $x$, $b$, and $c$ are column vectors. $\|\cdot\|_1$ is the L1 norm.
I have been reading about alternating direction method of multipliers (ADMM), but am not sure if this is the same setting. In particular, not sure if the constrained objective is convex or how to incorporate the nonnegative constraint. One technique could be to substitute $x$ with element-wise abs($x$) to enforce the nonegative constraint, but then is the resulting objective convex?
Any approach would be appreciated -- gradient method, alternating method, closed form solution, etc. Thank you.