Amortized and Rigorous Optimality Certificates for AC Optimal Power Flow via Load-Affine Dual Libraries

Proving an alternating-current optimal power flow solution globally ε-optimal usually takes a convex-relaxation solve per instance. Most of these solves can be reused. For a fixed multiplier, the box-constrained Lagrangian bound of a conic relaxation is affine in the load, so a multiplier computed once gives a lower bound at every load by an inner product. Where the relaxation's prices vary smoothly in few load directions, packing numbers in their number bound the size of a library from above and, under a curvature assumption, from below, and solving the relaxation only where the library falls short stays within the upper bound; where prices are constant within operating regimes, one multiplier per regime suffices. On eight benchmark systems of 14 to 500 buses, one multiplier per regime certifies nearly as many loads as a thousand, and a library used first makes an interior-point-plus-relaxation pipeline 3.0 to 15.2 times faster at ε = 0.5%, depending on the baseline configuration, and 8.9 times while proving both bounds. Grown this way over a year of hourly loads of a modified 2000-bus Texas grid, a library certified 98.2% of the hours with 252 solves.