  Many practical problems can be formulated as l0-minimization problems with
nonnegativity constraints, which seek the sparsest nonnegative solutions to
underdetermined linear systems. Recent study indicates that l1-minimization is
efficient for solving some classes of l0-minimization problems. From a
mathematical point of view, however, the understanding of the relationship
between l0- and l1-minimization remains incomplete. In this paper, we further
discuss several theoretical questions associated with these two problems. For
instance, how to completely characterize the uniqueness of least l1-norm
nonnegative solutions to a linear system, and is there any alternative matrix
property that is different from existing ones, and can fully characterize the
uniform recovery of K-sparse nonnegative vectors? We prove that the fundamental
strict complementarity theorem of linear programming can yield a necessary and
sufficient condition for a linear system to have a unique least l1-norm
nonnegative solution. This condition leads naturally to the so-called range
space property (RSP) and the `full-column-rank' property, which altogether
provide a broad understanding of the relationship between l0- and
l1-minimization. Motivated by these results, we introduce the concept of the
`RSP of order K' that turns out to be a full characterization of the uniform
recovery of K-sparse nonnegative vectors. This concept also enables us to
develop certain conditions for the non-uniform recovery of sparse nonnegative
vectors via the so-called weak range space property.
