From Approximate to Vertices: Set Distribution Matching for Linear Programming Crossover

First-order solvers such as PDLP substantially accelerate large-scale linear programming (LP). However, crossover procedures developed for interior-point solutions perform poorly on PDLP outputs, leaving costly vertex and basis recovery for industrial applications such as simplex reoptimization and tableau-based cut generation. We propose Basis-SDM to learn basis recovery from continuous solver states. We identify three challenges overlooked by variable-label supervision: a basis is an unordered set, its columns are interdependent, and degeneracy and multiple optima permit many correct bases. Basis-SDM addresses these challenges through a distribution over complete bases. It assigns order-invariant probabilities to sets; models column interactions through a determinant factor that excludes singular sets; and uses optimality certificates to define target distributions over compatible optimal bases. Shared updates move an initial distribution toward a target under exact KL supervision, followed by conditional-greedy basis selection for simplex cleanup. On 111 unseen LPs, Basis-SDM reduces cleanup iteration sGM from 367.99 to 43.66 relative to native COPT crossover with primal–dual hybrid gradient inputs, and from 518.52 to 4.74 with interior-point inputs. The KKT-adapted LP-GNN baseline obtains 274.40 on the latter, while Basis-SDM requires no cleanup on 64 instances. Controlled ablations demonstrate the benefit of supervising optimal-basis families. On 20 large MILPs, replacing native crossover in a hybrid GPU PDLP solver reduces mean complete solving time by 11.0%. Partial integer-assignment prediction further demonstrates the applicability of distributional supervision to other optimization decisions.