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Feature selection has been widely used in machine learning and data mining since it can alleviate the burden of the so-called curse of dimensionality of high-dimensional data. However, in previous works, researchers have designed feature selection methods with the assumption that all the information from a data set can be observed. In this paper, we propose unsupervised and supervised feature selection methods for use with incomplete data, further introducing an L_2,1 norm and a reconstruction error minimization method. Specifically, the proposed feature selection objective functions take advantage of an indicator matrix reflecting unobserved information in incomplete data sets, and we present pairwise constraints, minimizing the L_2,1-norm-robust loss functionand performing error reconstruction simultaneously. Furthermore, we derive two alternative iterative algorithms to effectively optimize the proposed objective functions and the convergence of the proposed algorithms is proven theoretically. Extensive experimental studies were performed on both real and synthetic incomplete data sets to demonstrate the performance of the proposed methods.