  The concept of data depth in non-parametric multivariate descriptive
statistics is the generalization of the univariate rank method to multivariate
data. Halfspace depth is a measure of data depth. Given a set S of points and a
point p, the halfspace depth (or rank) k of p is defined as the minimum number
of points of S contained in any closed halfspace with p on its boundary.
Computing halfspace depth is NP-hard, and it is equivalent to the Maximum
Feasible Subsystem problem. In this thesis a mixed integer program is
formulated with the big-M method for the halfspace depth problem. We suggest a
branch and cut algorithm. In this algorithm, Chinneck's heuristic algorithm is
used to find an upper bound and a related technique based on sensitivity
analysis is used for branching. Irreducible Infeasible Subsystem (IIS) hitting
set cuts are applied. We also suggest a binary search algorithm which may be
more stable numerically. The algorithms are implemented with the BCP framework
from the COIN-OR project.
