  We deal with monotonic regression of multivariate functions $f: Q \to
\mathbb{R}$ on a compact rectangular domain $Q$ in $\mathbb{R}^d$, where
monotonicity is understood in a generalized sense: as isotonicity in some
coordinate directions and antitonicity in some other coordinate directions. As
usual, the monotonic regression of a given function $f$ is the monotonic
function $f^*$ that has the smallest (weighted) mean-squared distance from $f$.
We establish a simple general approach to compute monotonic regression
functions: namely, we show that the monotonic regression $f^*$ of a given
function $f$ can be approximated arbitrarily well -- with simple bounds on the
approximation error in both the $2$-norm and the $\infty$-norm -- by the
monotonic regression $f_n^*$ of grid-constant functions $f_n$. We also
establish the continuity of the monotonic regression $f^*$ of a continuous
function $f$ along with an explicit averaging formula for $f^*$. And finally,
we deal with generalized monotonic regression where the mean-squared distance
from standard monotonic regression is replaced by more complex distance
measures which arise, for instance, in maximum smoothed likelihood estimation.
We will see that the solution of such generalized monotonic regression problems
is simply given by the standard monotonic regression $f^*$.
