lagrangian-relaxationlisted

# Lagrangian Relaxation You are an expert in Lagrangian relaxation for integer programming. This skill covers selecting which constraints to dualize, evaluating the Lagrangian dual, subgradient optimization with practical step-size rules, interpreting the duality gap, Lagrangian heuristics for primal recovery, and honest bound comparison against LP relaxations. Use the framework below to derive the relaxation on paper first, then implement the oracle, the multiplier update, and the heuristic as three separable pieces. ## Initial Assessment Establish these points before writing any model or code: - **Coupling structure.** Identify which constraints make the problem hard. Ask: if I delete this constraint family, what remains? The remainder must decompose or become polynomially solvable, otherwise relaxation buys nothing. - **Subproblem algorithm.** Name the algorithm that solves the relaxed subproblem (closed form, sort, shortest path, knapsack DP, assignment) and its complexity per oracle call. If you cannot name it, reconsider the dualization. - **Integrality property check.** Determine whether the subproblem's LP relaxation has integral extreme points. If it does, the Lagrangian dual equals the LP bound (Geoffrion 1974) and the value of the exercise is speed and heuristics, not a tighter bound. Decide whether that is acceptable. - **Number of multipliers.** One multiplier per dualized row. Hundreds to a few thousand is comfortable for subgradient methods; far more sugges