Diverse NMT aims at generating multiple diverse yet faithful translations given a source sentence. In this paper, we investigate a common shortcoming in existing diverse NMT studies: the model is usually trained with single reference, while expected to generate multiple candidate translations in inference. The discrepancy between training and inference enlarges the confidence variance and quality gap among candidate translations and thus hinders model performance. To deal with this defect, we propose a multi-candidate optimization framework for diverse NMT. Specifically, we define assessments to score the diversity and the quality of candidate translations during training, and optimize the diverse NMT model with two strategies based on reinforcement learning, namely hard constrained training and soft constrained training. We conduct experiments on NIST Chinese-English and WMT14 English-German translation tasks. The results illustrate that our framework is transparent to basic diverse NMT models, and universally makes better trade-off between diversity and quality. Our source codeis available at https://github.com/DeepLearnXMU/MultiCanOptim.