  In this paper we consider box constrained adaptations of $\ell_1$
optimization heuristic when applied for solving random linear systems. These
are typically employed when on top of being sparse the systems' solutions are
also known to be confined in a specific way to an interval on the real axis.
Two particular $\ell_1$ adaptations (to which we will refer as the
\emph{binary} $\ell_1$ and \emph{box} $\ell_1$) will be discussed in great
detail. Many of their properties will be addressed with a special emphasis on
the so-called phase transitions (PT) phenomena and the large deviation
principles (LDP). We will fully characterize these through two different
mathematical approaches, the first one that is purely probabilistic in nature
and the second one that connects to high-dimensional geometry. Of particular
interest we will find that for many fairly hard mathematical problems a
collection of pretty elegant characterizations of their final solutions will
turn out to exist.
