Abstract: The recovery of nonnegative sparse signals is not perfect given their underdetermined linear measurements at present, which can be improved further. This paper models this problem as linear programming and presents an optimization method with closed-form solution updated using the alternating direction method of multipliers. Moreover, the computational complexity is low. To enhance the sparsity of the recovered signal, this paper proposes the algorithm of iteratively reweighted linear programming, then the rate of successful recovery is increased by alternately optimizing the solution vector and the weighting vector. The effectiveness of the proposed algorithms and the recovering performance are verified by experiments on randomly generated signals and the actual power spectrum of speech signals, which outperform some state of the art sparse recovery algorithms.