Hyperspectral unmixing aims to extract the endmember and abundance features in an image. A hyperspectral image has many mixed pixels because of the low spatial resolution. Therefore, capturing the spectral features and the corresponding spatial distribution from the mixed pixels is important. The non-negative matrix factorization(NMF)-based method for hyperspectral unmixing is regarded as an ill-posed data-fitting problem, in which the cube data must be converted into a matrix form, which leads to the loss of three-dimensional structure information. This study introduces the sparsity of the spatial features in the minimum-volume simplex to propose a novel method for hyperspectral unmixing, which not only mines the intrinsic relationship between spectral and spatial abundance features in the images, but also improves the loss of data structure information. The proximal alternating optimization and the alternating direction method of multipliers were used here to design a set of efficient solvers based on the minimum volume constraint in convex geometry and non-negative matrix decomposition. After testing the synthesized and real data sets, the experimental results show that the proposed algorithm can effectively extract the endmember and abundance features.