Lagrangian relaxation

A Lagrangian relaxation moves selected constraints into the objective with appropriately signed Lagrange multipliers. Maximizing the relaxed objective over a larger easy set gives an upper bound for a constrained maximization problem. Optimizing that bound need not solve the original integer problem.

For binary knapsack optimization, $U(\lambda )$ is an upper bound for every $\lambda \ge 0$. Its minimum is computable from the finitely many profit-to-weight breakpoints and agrees with the fractional knapsack problem optimum. Zero-weight positive-profit terms are added independently.