  The optimization problem that arises out of the least median of squared
residuals method in linear regression is analyzed. To simplify the analysis,
the problem is replaced by an equivalent one of minimizing the median of
absolute residuals. A useful representation of the last problem is given to
examine properties of the objective function and estimate the number of its
local minima. It is shown that the exact number of local minima is equal to $
{p+\lfloor (n-1)/2 \rfloor \choose{p}} $, where $ p $ is the dimension of the
regression model and $ n $ is the number of observations. As applications of
the results, three algorithms are also outlined.
