National-Scale Manure Life-Cycle Optimization with Learned Prices and Optimality Bounds

National cattle-manure life-cycle optimization coordinates processing facilities, transport and crop application across roughly 33 billion potential transport variables. Restricting routes makes planning tractable but leaves the quality of the resulting plan unknown relative to the full model. We compute a feasible plan on a sparse network and bound the full optimum through a spatial relaxation that preserves local balances, transport and integer facility decisions. Lagrangian multipliers price shipments by material and destination, allowing independent regional solves whose upper bounds sum to a valid national bound. We learn eight material-specific adjustments to linear-programming dual prices from regional subproblem feedback and use them to initialize convex multiplier refinement. On 63,873 road-covered cells of the contiguous US, our approach produces a feasible $129.870B/yearplan,4.00$124.873B/year), and a $132.154B/yearupperbound,leavinga1.73$135.110B/year. On sixteen 256-cell regions from the same US data, learned initialization improves bounds in 15 cases after twelve rounds of the same optimizer, by a mean $0.570M/yearperregionrelativetoinitializationwithoutlearnedadjustments.Ourupperboundsarealso$9.018M/year lower on average than those from a recent learning-based approach to Lagrangian multiplier prediction.