Towards Solving Polynomial-Objective Integer Programming with Hypergraph Neural Networks

Bilevel mixed-integer linear optimization problems model hierarchical decision processes in which a leader anticipates the optimal response of a follower. Although expressive, these problems are computationally challenging because lower-level optimality is embedded in the leader's feasible region. Value-function reformulations replace the nested follower optimization with a constraint involving the follower's optimal value, but evaluating this value function exactly can itself be expensive. This paper introduces Graph4BiLO, a graph neural network (GNN) approach for learning bilevel value functions from variable--constraint graph representations. In contrast to fixed-length multilayer perceptron (MLP) representations, the GNN uses shared message-passing parameters and can therefore be applied across multiple problem sizes with a single trained model. The learned ReLU network is encoded exactly as mixed-integer linear constraints and embedded in an approximate single-level formulation. A repair step subsequently re-solves the follower problem for the selected leader decision to recover a bilevel-feasible follower response. We evaluate Graph4BiLO on knapsack interdiction instances with 20--100 items against the exact MibS solver and the learning-based Neur2BiLO method. Graph4BiLO obtains objective values comparable to Neur2BiLO across all tested sizes while avoiding size-specific neural networks. An additional out-of-distribution experiment demonstrates zero-shot transfer from 20-item training instances to previously unseen 40- and 60-item instances. However, embedding message passing at every graph node substantially increases the resulting mixed-integer formulation size and solve time. These results identify a central tradeoff between size-generalizable graph representations and the computational cost of embedding GNNs within optimization models.

Mixed-integer linear programming (MILP) is a foundational framework for combinatorial optimization across science and engineering, but remains hard to solve at scale due to NP-hardness. Recent learning-based methods typically model MILP instances as variable-constraint bipartite graphs and use Graph Neural Networks (GNNs) for representation learning, yet their locality limits representation power. We propose an attention-driven neural backbone that adopts an element-centric view of variables and constraints, with dual attention performing parallel intra-type self-attention and inter-type cross-attention. Across three representative tasks at the instance, element, and solving-state levels, our model consistently outperforms conventional GNN-based architectures, highlighting attention-based, element-centric modeling as a powerful foundation for learning-enhanced combinatorial optimization.

We introduce an autoregressive differentiable method to solve 0-1 integer programs. We fix an arbitrary order of the binary variables and we train a transformer to predict the next bit while remaining in the feasible set. Our method is first trained on feasible incumbents provided by any solver, thus allowing us to initialize the transformer in the feasible set. Our procedure then implements a Lagrangian penalty to penalize infeasible solutions, and the transformer is further trained to explore the feasible set using Gumbel-softmax activations on the relaxed objective. We have tested our method on non-convex instances of quadratic knapsack problem and demonstrated consistent improvement upon state-of-the-art open-source solvers for dense problems up to 10,000 binary variables. In particular, we empirically demonstrate a phenomenon akin to a tunneling effect where the effective change of variables from binary variable to the continuous weights of the transformer that the method implements enables crossing barriers in the relaxed objective landscape.