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Many practical problems can be formulated as l0-minimization problems withnonnegativity constraints, which seek the sparsest nonnegative solutions tounderdetermined linear systems. Recent study indicates that l1-minimization isefficient for solving some classes of l0-minimization problems. From amathematical point of view, however, the understanding of the relationshipbetween l0- and l1-minimization remains incomplete. In this paper, we furtherdiscuss several theoretical questions associated with these two problems. Forinstance, how to completely characterize the uniqueness of least l1-normnonnegative solutions to a linear system, and is there any alternative matrixproperty that is different from existing ones, and can fully characterize theuniform recovery of K-sparse nonnegative vectors? We prove that the fundamentalstrict complementarity theorem of linear programming can yield a necessary andsufficient condition for a linear system to have a unique least l1-normnonnegative solution. This condition leads naturally to the so-called rangespace property (RSP) and the `full-column-rank' property, which altogetherprovide a broad understanding of the relationship between l0- andl1-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 uniformrecovery of K-sparse nonnegative vectors. This concept also enables us todevelop certain conditions for the non-uniform recovery of sparse nonnegativevectors via the so-called weak range space property.

Equivalence and Strong Equivalence between Sparsest and Least $\ell_1$-Norm Nonnegative Solutions of Linear Systems and Their Application