TITLE: minimizing the sum of weighted absolute distance
QUESTION [7 upvotes]: Let $x_1, \ldots, x_n \in \mathbb{R}^d$ denote $n$ points in $d$-dimensional Euclidean space, and $w_1, \ldots, w_n \in \mathbb{R}_{\geq 0}$ any non-negative weights.
$\arg\min_{\mu \in \mathbb{R}^d} \sum_{i=1}^n w_i | x_i-\mu| = median\{x_1, \ldots, x_n \}?$
I understand why the median of $\{x_1, \ldots, x_n \}$ minimizes the function without the weights. But I am not sure if the median is what minimizes the function with the weights, should I use Lagrange multiplier to solve this?

REPLY [2 votes]: The problem you have stated is the weighted L1 median (sometimes called the weighted geometric median) of your data.  There are many references on the topic: 
https://www.google.com/search?q=l1+median
A concise reference along with an algorithm is:
http://www.stat.rutgers.edu/home/cunhui/papers/39.pdf
However, note that the Vardi-Zhang process needs to be adjusted for data having repeated values.  If you scan the literature, there are some references that describe this adjustment.