An Expandable Machine Learning-Optimization Framework to Sequential Decision-Making

We present an integrated prediction-optimization (PredOpt) framework to efficiently solve sequential decision-making problems by predicting the values of binary decision variables in an optimal solution. We address the key issues of sequential dependence, infeasibility, and generalization in machine learning (ML) to make predictions for optimal solutions to combinatorial problems. The sequential nature of the combinatorial optimization problems considered is captured with recurrent neural networks and a sliding-attention window. We integrate an attention-based encoder-decoder neural network architecture with an infeasibility-elimination and generalization framework to learn high-quality feasible solutions to time-dependent optimization problems. In this framework, the required level of predictions is optimized to eliminate the infeasibility of the ML predictions. These predictions are then fixed in mixed-integer programming (MIP) problems to solve them quickly with the aid of a commercial solver. We demonstrate our approach to tackling the two well-known dynamic NP-Hard optimization problems: multi-item capacitated lot-sizing (MCLSP) and multi-dimensional knapsack (MSMK). Our results show that models trained on shorter and smaller-dimensional instances can be successfully used to predict longer and larger-dimensional problems. The solution time can be reduced by three orders of magnitude with an average optimality gap below 0.1%. We compare PredOpt with various specially designed heuristics and show that our framework outperforms them. PredOpt can be advantageous for solving dynamic MIP problems that need to be solved instantly and repetitively.

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.