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The objective of single-channel source separation is to accurately recover source signals from mixtures. Non-negative matrix factorization (NMF) is a popular approach for this task, yet previous NMF approaches have not optimized directly this objective, despite some efforts in this direction. Our paper introduces discriminative training of the NMF basis functions such that, given the coefficients obtained on a mixture, a desired source is optimally recovered. We approach this optimization by generalizing the model to have separate analysis and reconstruction basis functions. This generalization frees us to optimize reconstruction objectives that incorporate the filtering step and SNR performance criteria. A novel multiplicative update algorithm is presented for the optimization of the reconstruction basis functions according to the proposed discriminative objective functions. Results on the 2nd CHiME Speech Separation and Recognition Challenge task indicate significant gains in source-to-distortion ratio with respect to sparse NMF, exemplar-based NMF, as well as a previously proposed discriminative NMF criterion.