A Tolerance-Aware Pricing Certificate for Stabilized Dantzig–Wolfe Branch-and-Price in Unit Commitment

Abstract

Stabilized proposal pricing may reduce exact subproblem work, but it cannot by itself certify branch-and-price node termination. A conditional, tolerance-adjusted node-pricing bound is developed for deterministic unit commitment. At the unstabilized restricted-master dual, CSDW-S applies a Phase-II cascade comprising a period-separable operational-envelope bound (Tier E), a chronology-preserving ramp-relaxed dynamic-programming bound (Tier D), and exact mixed-integer-programming pricing fallback. Relaxed optimizers are proof objects rather than master columns; CSDW-X prices every Phase-II unit exactly. The mechanism was evaluated in a prespecified 79-task study with a 600 s scientific deadline per task. In the primary 24-pair, 24 h horizon set, accepted root endpoints were returned by CSDW-S and CSDW-X in 15 and 24 conditions, respectively. Across the 15 jointly accepted pairs, the maximum absolute bound difference was 2.24×10−8. Exact true-dual unit solves were reduced by CSDW-S in all 15 pairs (median paired difference, −29,228), although CSDW-S was slower in 14 pairs (median paired difference, +20.7 s). The previously submitted 10-cell assessment did not conform to the primary acceptance criteria. It is therefore retained separately as historical evidence, with completion flags of 0/10 for CSDW-S and 9/10 for CSDW-X, and is not pooled with the new study. Mechanism, tolerance, horizon stress, and two small full tree contrasts support a qualified interpretation. Exact pricing workload can be reduced by the selective cascade when an endpoint is completed.