Crop yield prediction is crucial for global food security yet notoriously challenging due to multitudinous factors that jointly determine the yield, including genotype, environment, management, and their complex interactions. Integrating the power of optimization, machine learning, and agronomic insight, we present a new predictive model (referred to as the interaction regression model) for crop yield prediction, which has three salient properties. First, it achieved a relative root mean square error of 8% or less in three Midwest states (Illinois, Indiana, and Iowa) in the US for both corn and soybean yield prediction, outperforming state-of-the-art machine learning algorithms. Second, it identified about a dozen environment by management interactions for corn and soybean yield, some of which are consistent with conventional agronomic knowledge whereas some others interactions require additional analysis or experiment to prove or disprove. Third, it quantitatively dissected crop yield into contributions from weather, soil, management, and their interactions, allowing agronomists to pinpoint the factors that favorably or unfavorably affect the yield of a given location under a given weather and management scenario. The most significant contribution of the new prediction model is its capability to produce accurate prediction and explainable insights simultaneously. This was achieved by training the algorithm to select features and interactions that are spatially and temporally robust to balance prediction accuracy for the training data and generalizability to the test data.