Conformal Prediction for Early Stopping in Mixed Integer Optimization

Mixed-integer optimization solvers often find optimal solutions early in the search, yet spend the majority of computation time proving optimality. We exploit this by learning when to terminate solvers early on distributions of similar problem instances. Our method trains a neural network to estimate the true optimality gap from the solver state, then uses conformal prediction to calibrate a stopping threshold with rigorous probabilistic guarantees on solution quality. On five problem families from the distributional MIPLIB library, our method reduces solve time by over 60% while guaranteeing 0.1%- optimal solutions with 95% probability

💡 Research Summary

Mixed‑integer programming (MIP) solvers typically discover a high‑quality feasible solution early in the branch‑and‑bound process, yet they spend the majority of their runtime proving optimality. This paper proposes a fundamentally different strategy: instead of accelerating the proof of optimality, it learns when to stop the solver while still guaranteeing that the returned solution is within a user‑specified optimality gap ε with high probability.

The authors formalize a parametric family of MIPs, θ ∼ ϑ, and denote by Uθ(t) and Lθ(t) the solver’s upper and lower bounds at time t. The true optimality gap is gθ(t) = (Uθ(t) − z⋆(θ))/|z⋆(θ)|, which is unobservable during solving but can be computed after the run. Standard solvers stop when the algorithmic gap g_algθ(t) = (Uθ(t) − Lθ(t))/|Lθ(t)| falls below ε. Because gθ(t) often reaches ε far earlier than g_algθ(t), there is room for earlier termination.

The core of the method consists of two components. First, a neural network (implemented as an LSTM) ingests a rich solver‑state vector Xθ(t) containing the current bounds, best feasible solution, number of explored nodes, problem parameters, and elapsed time. The network outputs a raw score h(Xθ(t)), which is passed through a squashing function ϕ(x|l,u) = (u−l)·e^x/(1+e^x) to produce a predicted gap ĝθ(t) that is guaranteed to lie in