Data-driven prediction of relevant scenarios for robust combinatorial optimization

We study iterative constraint and variable generation methods for (two-stage) robust combinatorial optimization problems with discrete uncertainty. The goal of this work is to find a set of starting scenarios that provides strong lower bounds early in the process. To this end we define the Relevant Scenario Recognition Problem (RSRP) which finds the optimal choice of scenarios which maximizes the corresponding objective value. We show for classical and two-stage robust optimization that this problem can be solved in polynomial time if the number of selected scenarios is constant and NP-hard if it is part of the input. Furthermore, we derive a linear mixed-integer programming formulation for the problem in both cases.
Since solving the RSRP is not possible in reasonable time, we propose a machine-learning-based heuristic to determine a good set of starting scenarios. To this end, we design a set of dimension-independent features, and train a Random Forest Classifier on already solved small-dimensional instances of the problem. Our experiments show that our method is able to improve the solution process even for larger instances than contained in the training set, and that predicting even a small number of good starting scenarios can considerably reduce the optimality gap. Additionally, our method provides a feature importance score which can give new insights into the role of scenario properties in robust optimization.