  Given a dataset an outlier can be defined as an observation that it is
unlikely to follow the statistical properties of the majority of the data.
Computation of the location estimate of is fundamental in data analysis, and it
is well known in statistics that classical methods, such as taking the sample
average, can be greatly affected by the presence of outliers in the data. Using
the median instead of the mean can partially resolve this issue but not
completely. For the univariate case, a robust version of the median is the
Least Trimmed Absolute Deviation (LTAD) robust estimator introduced
in~\cite{Tableman1994}, which has desirable asymptotic properties such as
robustness, consistently, high breakdown and normality. There are different
generalizations of the LTAD for multivariate data, depending on the choice of
norm. In~\cite{ChaPitZiou:2015} we present such a generalization using the
Euclidean norm and propose a solution technique for the resulting combinatorial
optimization problem, based on a necessary condition, that results in a highly
convergent local search algorithm. In this subsequent work we use the $L^1$
norm to generalize the LTAD to higher dimensions, and show that the resulting
mixed integer programming problem has an integral relaxation, after applying an
appropriate data transformation. Moreover, we utilize the structure of the
problem to show that the resulting LP's can be solved efficiently using a
subgradient optimization approach. The robust statistical properties of the
proposed estimator are verified by extensive computational results.
