Learning Optimal Solutions via an LSTM-Optimization Framework

In this study, we present a deep learning-optimization framework to tackle dynamic mixed-integer programs. Specifically, we develop a bidirectional Long Short Term Memory (LSTM) framework that can process information forward and backward in time to learn optimal solutions to sequential decision-making problems. We demonstrate our approach in predicting the optimal decisions for the single-item capacitated lot-sizing problem (CLSP), where a binary variable denotes whether to produce in a period or not. Due to the dynamic nature of the problem, the CLSP can be treated as a sequence labeling task where a recurrent neural network can capture the problem's temporal dynamics. Computational results show that our LSTM-Optimization (LSTM-Opt) framework significantly reduces the solution time of benchmark CLSP problems without much loss in feasibility and optimality. For example, the predictions at the 85\% level reduce the CPLEX solution time by a factor of 9 on average for over 240,000 test instances with an optimality gap of less than 0.05\% and 0.4\% infeasibility in the test set. Also, models trained using shorter planning horizons can successfully predict the optimal solution of the instances with longer planning horizons. For the hardest data set, the LSTM predictions at the 25\% level reduce the solution time of 70 CPU hours to less than 2 CPU minutes with an optimality gap of 0.8\% and without any infeasibility. The LSTM-Opt framework outperforms classical ML algorithms, such as the logistic regression and random forest, in terms of the solution quality, and exact approaches, such as the ($\ell$, S) and dynamic programming-based inequalities, with respect to the solution time improvement. Our machine learning approach could be beneficial in tackling sequential decision-making problems similar to CLSP, which need to be solved repetitively, frequently, and in a fast manner.