Recent hyperspectral imaging systems are constructed on the idea of compressive sensing for efficient acquisition. However, the traditional reconstruction model in compressive hyperspectral imaging has a high computational complexity. In this work, compressive hyperspectral imaging and unmixing are combined for hyperspectral reconstruction in a low-complexity scheme. The compressed hyperspectral measurements are acquired with a single pixel spectrometer. The reconstruction model is represented in a space of lower dimension named linear mixture model. Hyperspectral reconstruction is then formulated as a nonnegative matrix factorization problem with respect to the endmembers and abundances, bypassing high-complexity tasks involving the hyperspectral data cube itself. The nonnegative matrix factorization problem is solved by combining an alternating least-squares based estimation strategy with the alternating direction method of multipliers. The estimated performance of the proposed scheme is illustrated in experiments conducted on a simulated acquisition in real data outperforming in 3dB the state-of-the-art reconstruction algorithms.