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In multicriteria decision-making, a user seeks a set of non-dominated solutions to a (constrained) multiobjective optimization problem, the so-called Pareto frontier. In this work, we seek to bring a state-of-the-art method for exact multiobjective integer linear programming into the heuristic realm. We focus on binary decision diagrams (BDDs) which first construct a graph that represents all feasible solutions to the problem and then traverse the graph to extract the Pareto frontier. Because the Pareto frontier may be exponentially large, enumerating it over the BDD can be time-consuming. We explore how restricted BDDs, which have already been shown to be effective as heuristics for single-objective problems, can be adapted to multiobjective optimization through the use of machine learning (ML). MORBDD, our ML-based BDD sparsifier, first trains a binary classifier to eliminate BDD nodes that are unlikely to contribute to Pareto solutions, then post-processes the sparse BDD to ensure its connectivity via optimization. Experimental results on multiobjective knapsack problems show that MORBDD is highly effective at producing very small restricted BDDs with excellent approximation quality, outperforming width-limited restricted BDDs and the well-known evolutionary algorithm NSGA-II.

Bilevel optimization deals with nested problems in which a leader takes the first decision to minimize their objective function while accounting for a follower's best-response reaction. Constrained bilevel problems with integer variables are particularly notorious for their hardness. While exact solvers have been proposed for mixed-integer linear bilevel optimization, they tend to scale poorly with problem size and are hard to generalize to the non-linear case. On the other hand, problem-specific algorithms (exact and heuristic) are limited in scope. Under a data-driven setting in which similar instances of a bilevel problem are solved routinely, our proposed framework, Neur2BiLO, embeds a neural network approximation of the leader's or follower's value function, trained via supervised regression, into an easy-to-solve mixed-integer program. Neur2BiLO serves as a heuristic that produces high-quality solutions extremely fast for the bilevel knapsack interdiction problem, the "critical node game" from network security, a donor-recipient healthcare problem, and discrete network design from transportation planning. These problems are diverse in that they have linear or non-linear objectives/constraints and integer or mixed-integer variables, making Neur2BiLO unique in its versatility.